mechanics exercise

,. ~-- ... ,_

. _,..,..,.~ --.-.,.-, -

. .: __ .,,,,! __ , ,.

Mastering

-

f

This set of books is written to provide a V'.ide range of classified questions that reinforce students' understanding in Advanced Level Physics, in the form of multiple choice questions and structured questions. It consists of four books according to the new HKALE syllabus: Mechanics, Waves, Electricity and Magnetism, and Matters.

In each chapter, there are

Brief notes intended to provide basic concept and allow quick revision,

Worked examples which demon~trate the applications of important formula and problem solving technique

Multiple choice questions grouped according to their objectives V\ith each question testing a unique concept

Exercises graded according to difficulties covering all aspects of the course requirement at A/ AS Level

Exam-type qu~stions which are highly structured, containing elements of comprehension and data analysis on unfamiliar situations

The main goals of this book are to develop student's confidence, to increase their understanding of natural laws, and to motivate their interests in the field of Physics. The questions in this book are designed to develop the ability to solve problems, to construct logical arguments involving a series of steps, and to apply them to real-life situations.

An effort has been made to see that the data correspond to reality and the situations correspond to real objects, for instance, with realistic masses mO\ing with realistic speeds. Effort has also been made to reduce the complexity of numerical calculations so that each question involves very little amount of Mathematics. With reference to the new trending of learning Physics in which more stress are put on experiments, detailed diagrams are drawn and, in some questions, use of instrument and familiarity of electric circuits are tested. Fachlal recalls are eliminated from the questions as far as possible. Students aiming for excellent results should find this book indispensable.

Solutions to all questions are published in a separate book "Solutions to Mastering Advanced Physics" in four other volumes.

Raymond W.N. Chan

September 2005 in Hong Kong

- j:~7-:~is=:--==.--~=-.,.,;.?:-~~-,::

:;.:.-::~~;.~~. ;:..:~:ii~'"''-?"-;_~

-..:.: . .,.

~E~ ~- ";;;:~ ~~{.._

- .. .,. ... --:::-

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Answers

Statics .............................................. 1

Brief Notes Worked Examples Multiple Choice Questions Exercises Exam-type Questions

Kinematics


Dynamics


Projectile Motion Brief Notes Worked Examples Multiple Choice Questions Exercises Exam-type Questions

Circular Motion


Gravitation


Oscillation Brief Notes Worked Examples Multiple Choice Questions Exercises Exam-type Questions

7 10 17 23

25

25 28 30 38 42

45

45 50 56 82 92

101

101 104 106 115 118

121

121 123 126 142 147

153

153 156 161 173 180

185

185 189 192 208 213

220

Chapter 2 Kinematics -------- ~---- . --~ . .. ... - ~

Chapter 3 Dynamics ----------~---.. - -"'---

Chapter 4 Projectile Motion Chapter 5 Circular Motion

--------------- . - -~--~~~--------- Chapter 6 Gravitation

1

25

45

101

121

153

185

( I ,

CHAPTER

1 BRIEF NOTES

Mathematical Formula

Algebra

Statics

1 {oo,forx>O If x--+ 0 then - = , (1 + x)" ""'1 + nx __ x - oo, for x < 0 and (1-xr" ""'1+nx

y=mx+c ""~"""'-'--~-~.......i y=kx~ ...... :..1.'1, ... ~-----=-----.._j

y y Positive slope

Negative slope -.......___,

Exponential and logarithm If x=lO ' ,then y= logx log(x y) = logx+ logy log(x/ y) = logx- log y

- 2.72

logx" = nlogx

\' ,,.

y=e -~---'

1 \0/ope is 1101 ~era here!

0

y = kx" or

X

I lk

~J~Z .. ~.~fut.J~i kJ logy

slope= 11

log k logx

0

0

If x = e', then y =I n x ln (x y) = In x +In y ln(xl y) = lnx-ln v ln x" = nlnx

-0.37

0 I l k

y

Ne ter I011ch the x-uxis

c y\ ..,::~4:~ "'-.__'-------- X

0

,. y = 1- e" . ~-"~--~ J

-0.63

X

I l k

I'

Brief Notes

2 Section A

Error Treatment

Brief Notes

Mechanics

Trigonometry sin( -8) =-sin 8

sin 8 tan(-8) =-tan 8 = ---

cos8 sin(90 8) = cos 8 sin 28 = 2 sin 8 cos 8

cos( -8) = cos 8

cos(90 8) =+sin 8 cos 28 = 2 cos 2 8- 1

sin(8) =sin 8coscos8sin cos( 8 ) = cos 8 cos+ sin 8 sin

Geometry For any triangle, c 2 =a 2 +b2 -2abcosC

a b c sin A sin B sin C

For a sphere, volume, F = 1m3 ; surface area, A= 4nr2

Function y=f(x) k kx

sin(kx) cos(kx)

e~cx

Function y=f(x) k

kx

x"

X

sin(kx)

cos(kx)

dy DifferentiaL

0 k

n-1 llX

11

dx

k cos(kx) -k sin(kx) ke"'

Integral, f y dx kx +c kx2 -+c

2 xn+1

--+c n+l

ln(x)+c

cos(kx) ----+c

k sin(kx) ---+c

k b

e -+c k

c

B~ c A

V'-/ Random error occurs when repeated measurements of the same quantity give rise to different values.

Systematic error refers to an effect that influences all measurements of a particular quantity equally.

Systematic errors may be due to zero errors

human errors failing instrument poor design of experiment Systematic errors affect the accuracy of the result

. ' '

r I f

t !

t l r

'

I

I t

I i

I j l I

I I

I f

i I

' .

Measuring instrument

Chapter 1 Statics

The largest probable error (uncertainty) is the maximum dev iation from the mean value. Percentaoe error = 6 A= largest probable error xI 00%

"' A mean value For A = x + y or A= x- y , M =I xI+ I y I . For A= cx; y" , where c and n are constan t, ~ =l~;'"l+n 16:1+1 6

2

2 1

Vernier Caliper

upper jaws (for internal diameter)

lower jaws (for external diameter)

sliding scale fixed scale

probe (for depth)

T:~p2 em 1111l1ffitrnir11

sliding scale - ~ I .1 step 3 I

0.1 mm ~

step 4 21+0.1=21.1 mm ~ _______ ..

Micrometer Screw Gauge

sleeve (rotatable)

step 4 1.5+0.33=1.83 mm J 1.5 mm

Brief Notes

3

4 Section A Mechanics

Free A~t~r~e~s~t------------------------------, body diagram

Acceleratin

Concurrent forces

Toppling of object

Moment of forces

Static Equilibrium

Brief Notes

Condition for equilibrium: For an object acted on by two or more forces in different directions,

the net force is zero: L F = 0 the external forces are concurrent with the weight (i.e. passing through the same point)

Toppling begins to occur when the normal reaction reaches the lower edge of the object.

Moment of force measures the turning effect of the force.

1 = Fxrsin 8 I

The general conditions for static equilibrium: I. The net force on the object is zero: L F = 0 2. The total moment about any point is zero L 1 = 0

toppling begins to occur

\

r r

:

r I

' I

r

Vector nature of forces

Center of gravity

Stability

Friction

Resultant of two forces

Components: {Horizontal: Vertical:

F, = F; cos 8 I + F2 cos 82 FY = F; sin 81 + F2 sin 82

Chapter 1 Statics

Resultant F

Resultant: Direction: 8 = tan_, ( ~ J Rod

,~';':','~,I w I~ It' I I It I C. G.

x=-'-;=:::'--- or x = -";=:::'---w M

w, W, w,

x, X, X X,

Lamina

Applied force

I. 2. 3.

x=-'-'=:::'---M

N

2)m;y;) y =-'-'=-'_, __ _

M

C.G may be located outside the object. (e.g . L-shaped objects) C.G can be found by suspending the object freely at two points. It is~e-i~ecting point of lines drawn using a~lumb-lin2\

0

Neutral equilibrium Stable equilibrium Unstable equilibrium

When displaced, the C.G remains at the same level When displaced, the C.G is raised. When dt splaced, the C. G. is !oweted.

N eutral Stable Unstable equilibrium equilibrium equilibrium

IV

.r-axis

Friction occurs between two contact surfaces in a way to resist their relative motions. Lubrication or the use of bearings can reduce kinetic friction, and helps to energy savtng.

Brief Notes

5

6 Section A

Kinetic friction

Static friction

Coefficient of friction

Friction in a car

Hooke's Law

Combination of springs

Brief Notes

Mechanics

Kin etic friction fk usually takes on a fix ed value, independent of the relative speed of the contact surfaces.

It accounts f()r !Tl91?L~_!)_ergy dissipation. For instance, it occ~~ between the rotating shaft of a car and its chassis.

Static friction f, is a lways equal to the applied force, as there is no rel ative motion between the contact surfaces.

Static friction will not cause energy di ssipation. The limiting static friction f L is independen_t_cif _the .

;

' i ' t

I I I !

f l I

..

Chapterl Statics

WORKED EXAMPlES

EXAMPLE 1

SOLUTION

EXAMPLE 2

A workman uses a set of ropes to unload a l 200 N cargo from a truck as shown. The ropes are arranged so that the segment pulled by the workman is horizontal. As the mass is raised a small height from the truck, find

(a) the tension Tin the upper rope (b) the force F supp lied by the workman

T

F

1200 N

Fig. a Fig.h

Consider the forces at the junction of the ropes (see Fig. b ). At equilibrium , the vertical and horizontal forces are balanced. We have

{Tcos20 = 1200 ...... .. .. (!) F = T sin 20 ..... .. ... (2)

From ( I) , T = 1277 N = 1300 N Puttin g into (2), F = 437 N = 440 N

Two identical smooth spheres of weight l N and radius rare placed inside a smooth cylinder of radius l.Sr. Find

(a) the force N between the sp heres, (b) the forceR acting on the upper sphere by the cylinder.

1.5r ~ Forces on the upper sphere N

R\ R e

w

Fig.a Figb Fig.c

Worked Examples

7


SOLUTION

EXAMPLE 3

SOLUTION

.):: From the geometry in Fig.a , the angle 8 that the lin e joining the centers makes with th e horizontal is given by

cos 8 = __::__ . . 8 = 60 2r

As the surfaces are smooth, the contact force s are at right angle to the surfaces of the spheres, i.e. they are radial (see Fig.b). Consider the forces on the upper sphere (see Fig.c). The x- and y-co mponen ts are respectively

..... (!) {

N cos8 = R Nsin8 = W .... ... ... .. ... (2)

IV 1 From (2), N = -- = -- = 1.15 N sin 8 sin 60"

Putting into (I), R = __!!_ = ~ = 2.31 N cos 8 cos60o

A stage is set up by a wooden plank PQ of weight 1000 N and length 12 m . The stage is supported by two ropes, A and 8, which are both 2 m from the ends. A workman of weight 600 N walks from rope 8 towards end P.

A B

p 600 N Q

Fig. a 12m

(a) Describe the variation in tension of the rope. (b) State whether the stage will topple before the workman can reach P?

(a) Let x be the distance of the workman from rope A (see Fig. b). Initially, x = 8 m. At P, x =-2m.

T' I X I -- l! T p 1- 4m Q

A 1000 N l t B Fig.b 600 N

Taking mome nt about the point where rope A is attached to the plank,

4 x 1000 + 600x = 8T T = 75x + 500 ....... .. (])

As the workman moves towards P, x decreases from 8 m to-2m, i.e. T always decreases. On the other hand , the tension T' in the other rope must increase because the weight is balanced by T and T'. Therefore, during the movement, tension in rope A increases and that in rope B decreases .

.

(b) For th e plank to topple, one of the ropes slacks. Putting T = 0 into (I) ,

0 = 75x + 500 :. x = -6.67 m

Thi s is imposs ible sin ce th e sma lles t value of xis -2m.

Worked Examples

. ' \

' i

r [._

r--r-

EXAMPLE 4

Chapter 1 Statics

A uniform ladder of length 4 m and weight 1 000 N leans against a wall making an angle of 60 to the horizontal. The maximum friction between the ladder and the ground is 400 N.

(a) Determine the normal reaction on the ladder from the wall. (b) Hence, find the friction between the ladder and the ground. (c) A man of weight 500 N climbs up the ladder. How far can the man climb up the

ladder without causing the ladder to fall down?

-- smooth wall

;,;~~~ ' :i! -:.::~~

~~** . _..t.,_ . -...:O.r rough ground

.\"

R' R 60

p -- ' p -----~-- ~ .. ....,_ ~ ~

.. t ;~??!lC~.:;: / J.: .... = 400 N

SOLUTION

Fig. a Fig.b Fig.c

(a) Fig.b shows all the forces acting on the ladder. T aki ng moment about th e lowest point P. the normal reac tion N from th e wall is give n by

N x 4si n 60 = 1000 x 2cos60" N = 289 N

(b) The gro und exerts two forces on the ladder: the friction f to the left and the normal reaction R upward. As hor izonta l force are balanced, f = N = 239 N .

(c) The ladder falls if the friction between the ladder and the ground is not great e nou gh to balance the no rm al reac ti on N' from the wal l (see Fig.c). The limiting case occurs when

N' = J.na< = 400 N .

Refer to F ig.c. Taking moment about P,

N' x 4sin 60 = 1000 x 2cos6o + 500 x cos6o 400 x 3.46 = 1000 + 250x

x = 1.54 m

Worked Examples

9


MUlTIPlE CHOICE QUESTIONS

Error Treatment

l.

2.

In an experiment to determine the acceleration due to gravity , a metal ball is dropped from rest through two pairs of metal foils as shown. The timer scaler is used to measure the time interval in which the ball moves between the frames. Which of the foll owing may reduce the random error?

Metal foil 0

(I) Repeat the experiment for many times. (2) Use a larger metal ball so that its radius

can be measured more accuratel y. (3) Release the ball closer to the upper frame.

A. (l)only B. (3) only C. (1) and (2) only D. (2) and (3) only

An experiment is repeated for several times. Given that the standard value of a quantity is x0 The charts below represent the distribution of readings in four experiments. N represents the number of occurrence of a given reading x.

Which experimental result is neither precise nor inaccurate?

A. N ...

X () L_ __ ~LL~~LL~--~~

x ..

Multiple-Choice Questions

3.

4.

B. N

o~---.l.....l....l..;..l_J_J __ _..

x ..

c. A'

X ()

x ..

D. N

n X

0 .\

In an experiment to measure th e thickness of a metal tube, the external diameter and internal diameter are found to be (82 I ) mm and (76 1) mm respectively. What is the best representation of the thickness of the tube?

A. 32mm I B. 3 1 nun

L)

c. 61mm D. 62mm

A student measures the diameter d of a sphere to determine the volume V. If the percent error in d is 5%, what is the percent error in V?

A. 5% B. 10% c. 15% D. 50% .

.,

5.

6.

7.

8.

A micrometer screw gauge is used to measure the diameter of a piece of wire. The following readings were obtained :

mean zero reading -0.15 0.02 mm, and mean diameter +3.25 0.02 mm.

The diameter of the wire should be written as

A. 3.10 0.02 mm B. 3.10 0.04 mm C. 3.40 0.02 mm D. 3.40 0.04 mm

m v2

The formula F = -- is used to calculate the l

centripetal force. If the percent errors in m, v and l are 2%, 3% and 4% respectively, the percent error in F is

A. 4% B. 5% c. 9% D. 12%

In an experiment to determine the area of cross-section of a metal wire, a student measures its diameter and obtains a value of 0.80 mm, subject to an error of 0.04 mm. Which of the following is the most appropriate way of expressing the result ?

A. 0.5026 0.025 mm2 ' ' B. 0.50 0.03 mm2 C. 0.5026 0.05 mm2 D. 0.50 0.05 mm2

In an experiment to meas ure the pressure due to the weight of metal cube, the following measurements were obtained:

weight of the cube = 132 3 N side of the cube= 0.034 0.001 m

Estimate the percent error in calculating the force

value of pressure using the formula p = -- . area

A. 2% B. 5% c. 8% D. 11%

9.

Chapter 1 Statics

The period of oscillation, T, of a si mple pendulum is related to its length , l, by the

formula T = 2:rc J7i . To find experimentally the acceleration of free fall by using in simple pendulum, a student takes the following measurements:

time for 10 oscillations : 13.8 0.2 s length of the pendulum : 0.472 0.001 m

Which of the following is the most appropriate way of expressing the result?

A. 9.8 0.3 m s-2 B. 9.78 0.304 m s-2 C. 9.8 0.2 m s-2 D. 9.785 0.163 m s-2

10. Which of the following experimental techniques reduces the systematic error of the quantity being investigated?

A. timing a large number of oscillations to find a period.

B. measuring the diameter of a wire repeatedly and calculating the average.

C. adjusting an ammeter to remove its ze ro error before measuring a current.

D. using a metre rule graduated in 0.5 mm rather than I mm to meas ure diameter.

11. In an experiment to determine the period T of an oscillation, the time, t, for a number of complete oscillations is taken. It is found that the time for 20 complete oscillations is 36.5 0.2 s. Which of the following statements is/are correct ?

(l) The reading error in t can be reduced by counting 100 oscillations.

(2) The percent error in T is the same as that in t.

(3) The period T determined should be quoted as 1.83 0.01 s.

A. (l) only B. (3) only C. (1) and (2) only D. (2) and (3) only


1 1


Force as a Vector

12. If a force of 10 N is resolved into two perpendicular components, which of the following statements is/are ROSsible?

(1) One component is 6 N; the other is 8 N. (2) Each component equals 5 N. (3) One of the components is greater than 10

N.

A. (1) only B. (3) only C. (1) and (2) only D. (2) and (3) only

13. The resultant of two forces F 1 and F2 acting at a point can have a maximum value of 7 N. When the two forces act at right angles to each other, the magnitude of their resultant is 5 N.

What is the minimum resultant of these two forces?

A. 0 B. 1 N C. 2N D. 3 N

14. A ball suspended by a string can be balanced by an external force in two cases as shown above. F is applied horizontally and F' is applied at right angle to the string.

Case A Case B

What is the difference in size between the two forces?

A. 0 B. nzg(tanG-cosG) C. mg (cos B- sin B) D. mg (tanG- sin G)


15. The figure shows a toy plane with an air propeller connected by a light string to a fixed point P on the ceiling. The plane remains stationary when the propeller is on. .

p

air propeller

\ ~ toy plane

Which of the following diagrams correctly represents the forces acting on the plane?

A.

B.

c.

D.

,.

16. The directions of three forces acting on a body are represented by three arrows in each of the diagrams below. In the absence of any other forces, which of the bodies cannot be in static equilibrium under any circumstances?

(I)

(2)

(3)

A. ( 1) only B. (3)only C. (!) and (2) only D. (2) and (3) only

Pulley

17. A heavier mass /vf and a lighter mass m are connected by a light string passing over a smooth pulley of negligible mass. The system is kept stationary by a boy holding !VI.

.'vi Ill

Chapter 1 Statics

What is the force acting on the hand of the boy by M?

A. Mg upward B. Mg downward c. Mg -mg upward D. Mg-mg downward

18. A block of mass 100 kg is to be lowered stead ily at a~ s 1 by an applied force F at the end of a string passing over a fixed freely-runn ing pulley. Take g = 10m s-2

. ~9;C-r .......

smooth pulley

F

100 kg

A. F is less than 1000 N but greater than zero.

B. F is equal to 1000 N. C. F is g reater than 1000 N but less than

2000 N. D. F is greater than 2000 N.

19. The system of masses shown is in static ....__...., equilibrium.

T,

T,

50 kg 100 kg

Which of the following is correct?

A. Tt > T2 > T3 B. T2 > T, > T3 C. T3 > Tz > T1 D. T, > T2 = T3


13

14 Section A Mechanics --------'---'----- -~---- .

Friction

20. A slowly increasing horizontal force is applied to a block resting on a rough horizontal surface.

slowly increasing applied force

Which one of the following graphs correctly represents the relationship between the sizes of the applied force F and the frictional force f acting on the block?

A.

F

B.

F

c.

F

D. I

F o""-----'----_.

21. A block of wood of mass 1.0 kg is gently placed on an inclined plane which makes an angle of 60 to the horizontal. The coefficient of static friction between the block and the plane is 2.0 and the coefficient of kinetic friction is 1.5. What is the frictional force acting on the block? Take g =10m s2.

A 5.0N B. 7.5 N C. 8.7 N D. 10 N


22. Two blocks are connected by a light string passing over a light, frictionless pulley as shown. The largest frictional force between the 1.0 kg mass and the inclined plane is 4.0 N.

rough plane (max. friction 4 N)

111

Detennine the mass of m when the 1.0 kg mass is on the point of (a) sliding up and (b) sliding down the plane. Take g =10m s2 .

sliding up sliding down A 0.1 kg 0.9 kg B. 0.5 kg 0.5 kg c. 0.9 kg 0.1 kg D. 0.9 kg 0.9 kg

23. Two wooden blocks P and Q are connected by a string which passes over a smooth, fixed pulley as shown. The maximum friction between any two surfaces is 1 N.

F

If a horizontal force F is applied to block Q, find its minimum value for moving Q.

A IN B. 2N C. 3N D. 4N

Moment of force

24. The diagram shows a uniform rod of length .!JL!rl freely pivoted at P, suspended horizontally and at rest in the manner shown. M is a 3 kg mass attached to a weightless string that passes over a smooth pulley and tied to the end of the rod.

25.

I ~ I

p

lm

The mass of the rod is

A. 1.5 kg B. 3.0 kg c. 4.0 kg D. 6.9 kg

3 kg

A person climbs up a ladder PQ which is supported by a smooth wall and a rough ground.

Q -- smooth wall

p

rough ground

Which of the following would increase as the person climbs up the ladder?

(1) normal reaction at P (2) normal reaction at Q (3) friction at P

A. (1) only B. (3) only C. (2) and (3) only D. (1) , (2) and (3) only

26.

27.

' ,,

Chapter 1 Statics

A uniform metre rule of weight 2 N is pivoted at the 70 em mark as shown. A mass of weight 5 N is suspended at the 100 em end.

70 em --"i

pivot P

5N

When the rule is horizontal, what is the resultant turning moment about the pivot P?

A. 0.5Nm clockwise B. 0.5Nm anti-clockwise c. 1.1 Nm clockwise D. l.INm anti-clockwise

r X - - d 0

The torque produced by the pair of forces F about the point 0 as shown is

A. Fd clockwise B. Fd anti-clockwise C. Fx clockwise D. 2Fd clockwise


15


28__ For safety reasons, a vehicle should be so designed that no side ways toppling occurs before reaching an angle of inclination of 30.

If the centre of gravity of that vehicle is 1.0 m above the ground, what is the minimum separation h between its wheels?

A. 0.58 m B. 1.15 m c. 2.31 m D. 3.00 m

29. The figure shows a uniform rigid beam PQ, pivoted at P, held in horizontal position by n wire attached to a wall at point R, vertically above P. The beam carries a load W.

R

Q p

w

If W is shifted gradually from P towards Q, which of the following quantities will increase?

(I) The tension in the wire. (2) The horizontal compression force in the

beam. (3) The vertical component of the reaction P.

A. (I) only B. (3) only C. (1) and (2) only D. (2) and (3) only


30. A block rests on a rough inclined plane. In the following diagrams, the forces acting on the block are represented by

W : the weight of the block, f: the friction and R : the normal contact force by the plane.

Which of the following diagrams correctly shows the lines of application of these forces acting on the block?

A.

X B.

r

~ c.

~~ D.

ltJ ' l

l '7 I

Chapter 1 Statics

\

EXERCISES

In all calculations, take g = 10m s2 and ignore air resistance unless specified otherwise.

ERROR TREATMENT

1. The width and height of the screen of a notebook computer are quoted as 0.28 m and 0.17 m. State the area of the screen giving the result to the number of significant figures that are justified by the data.

2. The length l of a rod is measured. The reading is l = 23.5 0.2 em. What is the percentage error of the measurement?

3. A metal ball is allowed to fall through a column of viscous oiL The time of motion t Js repeatedly measured for six times and the result is given below:

12.8 s, 12.7 s, 13.0 s, 13.1 s, 12.6 s, 13.0 s

(a) Use your calculator to compute the mean value of the time of motion. Give your re sult corrected to one decimal place.

(b) Hence, find the probable error for the time of motion and write down the mean value again with the error taken into account.

4. The period of a simple pendulum of length I is T = 2nj!i If the percentage error in measuring Tis 0.1 % and in measuring I is 0.8 %, what is the percentage error in the calculated value of g, the acceleration due to grav ity?

5. In an experiment, the fol lowing data are taken:

a=20 1,b=12 2,c=50 1

Calculate the value of each of the fo llowing quantities, stating the uncertainty.

(a) X= 2a + 4b (b) Y = 6a b

(c) z = Sa3b

c

6. In an experiment to determine g, the acceleration due to gravity, a stone is dropped from a height h which is measured to be 9.05 0.01 m. The time of flight, t, is repeatedly measured for five times:

1.41 S, 1.45 S, 1.42 S, 1.43 S, 1.40 S

(a) Calculate the percentage errors in h and t.

(b) What is the value of g? Give your answer with the probable e rror.

Exercises

17

18

7.

Section A Mechanics

The mass of a ball bearing is measured to be 36.2 0.2 g and the radius is found to be 4.02 0.05 mm. Calculate

(a) the percentage error of each measurement,

(b) the percentage error of the density, and

(c) the value of density and its probable error.

Suggest an instrument used in measuring the radius of the ball bearing.

8. In an experiment to measure the specific heat capacity of a liquid, the liquid is heated in an insulated container using a 50 W heater. The following data were obtained:

mass of liquid temperature rise time for which heater runs

=so 1 g = 20 I K = 35.0 0.2 s

(a) Calculate the value of the specific heat capacity of the liquid.

(b) Identify which measurements has the greatest percentage error.

(c) Determine the absolute uncertainty in the value of the specific heat capacity.

9. A rectangular block has a mass of 1.2 0. I kg and its dimensions are: 20 2 mm. 25 2 mm and 30 2 mm. Calculate

(a) the percentage error in mass and volume,

(b) the density of the block, stating its absolute uncertainty.

10. The external diameter a and the internal diameter b of a metal tube are measured to be 50 0.5 mm and 48 0.5 mm respectively. What is the percentage error in finding the thickness of the metal tube?

The answer illustrates the need for careful measurement when dealing with the difference between two almost equal quantities. Suggest a method to improve the reliability of the result.

MIRCROMETER SCREW GAUGE & VERNIER SCALES

11. The following diagrams show a micrometer with a screw pitch of 0.5 mm. The circular scale has 5Q divisions on it. What is the reading in each case?

(a) (b)

(c) (d)

Exercises

t .

Chapter 1 S~atics 1 9

12. The following diagrams show a sca le graduated in centimetres and millimetres together with a millimetre vernier. What measurement is indicated in each diagram?

(a)

em 2 3 4

1111111111111111111111111111111 [1111111

(c)

em 9 10 11

111111111111111111 1 1 1111

111111 111111

~?.. o ::L~ft;1._: :i;_:~~~~:c .. J f ORCE, STATICS

(b)

4 5 6 em

11111111

1

11111111

1111 111111111111111

. 0

(d)

15 16 17 em

111111111111 11111 11 111 11111 1111 r 1111111 1 O~.::....~.L~_:L:.:: ... :_.~

13. A point object is acted on by forces 3.0 N, 4.0 N and 5.0 N and is in equilibrium. If the 5.0 N force is removed, what is the resultant force acting on the object?

14. A block on a smooth horizontal ground is pulled by two forces. 450 N due north and 370 N in a direction N50E. as shown. What is the direction of motion of the block?

15. Two forces 12 Nand 16 N are acting on a point object. Calculate the re sultant force if the lines of action of the forces are

(a) parallel but in opposite direction, (b) at right angles to one another, and (c) at 135 to one another.

16. Determine the coordinates of C.G. of the uniform L-shape lamina shown. -

( 17. 1Jwo identic.! blook mh of length L '" ""nged ,: hown. Find \,__/he greatest overhanging distance x of the upper block from the

table edge. (Hint: Consider equilibrium starting from the top to the bottom) ---------- -

L

4

3

2

0

}

450 N

50 '

370 N

2 3 4 X

Exer cises

20

I '

Section A Mechanics

18. A half-metre rule is held at one end in two different ways:

19.

20.

Fig. a Fig.h

(a) On Fig. a draw and label an arrow to represent the weight W of the half-metre rule and an arrow to represent the force F provided by the student's hand .

(b) In Fig.b, the rul e is held horizo ntally between the thumb and the first finger. i) Draw and label all the forces acting on the half-metre rule.

ii) List these forces in order of increasing magnitude .

A uniform metre rule is balanced horizontall y on a pivot at its 30 em mark by hanging a 14 N 1oocm weight on a thread from the zero mark. Calculate the weight of the rule.

r 15m Fig. a

metre rule

Fig.b

30 em

A truck, supported by three shafts, is at rest on a level ground as shown in Fig.a. The front shaft takes q,ne half of the total weight of t~d the rear shafts share the other half equally.

(a) Calculate the horizontal di stance of the centre of gra vity from the front shaft.

(b) If the truck is at rest on a slope as shown in Fig.b, describe qualitatively how force acting on the front shaft changes.

2 I. In the diagram shown, the kinetic fri ction between the blocks is 2 N and the kinetic frict ion between the lower block and the table is 5 N . What is the minimum force required to start the motion?

Exercises

..

' I

22. A heavy chain PQ is used to support an object as shown. The weights of the chain and the object are 10 N and 20 N respectively . Find the tension in the cha in

(a) at P,

(b) at Q, and

(c) at the middle of P and Q.

Chapter 1

p

Q

Statics

heavy chain ION

20 N

23. A man of mass 80 kg stands at the middle of a 4 m long uniform beam of mass 10 kg. The beam is supported at both ends by strings which break under a tension of 650 N. How far from the centre of the

2 1

beam can he move before one of the strings breaks? ..... ) ~ gao .A : c~o(J'-) i ) ~-"c ,n ' ~ ~ ....

A_ -:..1 1A --v :;. \~ 1' f 24. A block of mass 5 .0 kg is fixed to the end of a uniform

metal rod PQ hinged to a vertical wall at P. The rod is ~d has a mass of 1.0 kg. The rod is supported at Q by means of a rope which is fixed to the wall at R and makes an angle of 30 to the horizontal.

(a) Taking mo ment about P, find the ten sion in the rope. (b) Find the vertical compo nent of the tens ion in the rope. (c) Find the total downward force acting on the rod. (d) Exp lain why the values in (b) and (c) are different. (e) Hence, find the magnitude and direction of the force

ac tin g at end P of the rod by the hinge. ( f) Draw a diagram show ing all the forces act ing on the

rod .

25. The diagram shows a h_eavy_~k hinged to a ve rti ca l wall at one end and sup ported by a strut whi ch inclines at 60 to the vert ical. The we ight of the plank is 200 kN. Assume that the thrust from strut a~ts along its lengt!:!.: Calculate

(a) the thrust from the stru t, and

(b) size and direction of the force at the hinge.

26 . A pulley , hung from a f ixed support by a rope PQ , is used to raise stead il y a 200 N load as show n. The two portions of the rope make an angle of 60 with each other. Assume that the rope and the pulley are weightless and any effec t due to friction is negligible . Find

(a) the angle of PQ to the vert ica l and

(b) the tension in PQ.

e: ! r R .;, -. -~--~

I kg

p

plank

Exercises


27.

p

The diagram shows a 100 kg load being tran sported across a river by means of two ropes and a fixed pulley. The rope PQ passing over the pulley is maintained by persons on one side of the river. The other ropeRS is pulled by a truck on the opposite bank.

Ca lcu late the tension in each rope

(a) when the load is halfway across the river as shown,

(b) when the load has just reached the right bank.

28. A wheel of radius 1.0 m is to be li fted over a kerb of height 0 .50 m by applying a horizo ntal force of 50 N on the axle of radiu s 0.20 m.

29.

(a) Calculate the weight of the wheel.

(b) If the force is applied in some 'either di rection, the force needed can be reduced. Fin'd the minimum force required to lift the wheel.

t A system of three pulleys is employed to hold a 21 kg mass at equilibrium. There is no friction and the pulleys are weightless.

Calcu late

(a) the applied force F, and

(b) the tension Tin the upper cable.

Exercises

l ) - / '.

J A

F

I '

'- t

.~r. ~ ~ -,

""

Chapter 1 Statics

EXAM-TYPE QUESTIONS 1. All the pulleys in this q uestion a~

T, T

A B Q --~------ 12 em

Fig.a same weigh! Fig.b Fig.c

(a) Two identical blocks are suspended by a string and a pull ey in two different ways as shown in Fig .a and Fig.b. Compare

i) the tensions T 1 and T2 in the strings that pass over the p ull eys

ii) the tensions T' 1 and T'2 in the strings that suspend the pulleys. (2 marks)

(b) A plank PQ of length 12 em and weight 90 N is suspended at two ends by a system of pulleys as shown in Fig.c.

i) Determine the tension Tin the string that passes over the pulleys .

ii) Find the position of the center of gravity of the plank. Hence, state what happen to the system if the plank used is uniform. ( 4 marks)

(c) A workman of weight 600 N standing on a uniform plank of weigl~'\f is trying to keep himself ~1s shown in Fig.d. The plank is shown in do tted li ne-:-

i)

ii)

workman 600 N

Fig.d

plank 400 N

Assume that the workman arid the plank are in equilibrium. Find the force F that the workman must exert on the string, and the normal reaction R between the workman and the plank.

By drawing a free-body diagram of the plank, show that the p lank located in the dotted line wi ll turn about a horizontal axis. State the direction of the rotation and draw in Fig.d to show the correct position of the plank. (6 marks)

Exam-Type Questi o n s

23


2. (a) ln an experiment, a heavy uniform half-metre ule is supported horizontally by a pair of first fingers, with one finger near

CHAPTER

2 .; 4. t.t. :t

Kinematics

BRIEF NOTES

Speed

Velocity

Change in velocity

Relative velocity

Acceleration

Speed is a scalar quantity that indicates how rapid an object is moving.

- Distance travelled s ill Average speed, v = -------

Time taken t:lt

Instantaneous speed is the average speed when t:l t is very small

. (t:ls) ds Instantaneous speed, v = li m - =-6t->O Cl/ d (

Velocity is a vector quantity that describes the rate of change of displacements.

. - Displacement s ru Average velocity, v = _ _o_ __ _ Time taken t:lr

l . 1. (ru) ds 1 Instantaneous ve oc1ty, v = 1111 - =-ill->0 t:lt d [

When velocity changes from v1 to v2 , the change in veloc ity is found by vector subtraction

I t:lv = v2 =vi I ThevelocityofArelativetoBis I v,JJ=v,-v 8 i Acceleration is a vector quanti ty that desc ri bes the ra te of change of velocity v.

. Chancre in velocity Average acceleration =;--"""'----=-. Time taken

v - u t:lv a =--=-

t t:lt

Instantaneous acceleratiOn, a = hm - = -. . (t:lv) d v ill->O Cll d f

The sign of acceleration depends on which direction is taken as positive. Even a is pos itive, the speed may increase or decrease, depending on its current direction of motion. e.g. Consider an object projected upward. If downward is taken as pos itive, the acceleration is positive. The speed will decrease to zero, then increase.

Brie f Notes

26 Section A

Velocity-time graph Brief Notes

Mechanics

For uniform acceleration,

v=u+at

s = ut+ 1at 2

v2 = u

2 + 2a s u+l '

s=--1 2

Consider an object projected vertically upward from the ground (Neglecting air resi stance and taking upward as positive):

upward downward

1 /s 5m

0 2

.. 10 ms I t -10 ms' -1 0 - ------------,-- - --. ----- ...

The time for upward equal s th at for downward. The acceleration is the same for upward and downward (a= -g =-10m s-2). The speed of projection equals the speed of return. The net di splacement is zero, after returning.

Consider an object projected up a smooth inclined plane (Taking up-slope as positive): upward downward

v /ms' 1 0 ....................................... --------------.................. _ ............. .

0

t I !2

I l !

-1 0 -----------~--------------------- .

The acceleration is the same for upward and downward (a= -g sin 30 = -5 ms'2 ) For an inclined plane at angle 8 to the horizontal , the equation of motions are

v = u - g sin e . t S = U I -1 g sin 8 12

v2

= u2

- 2 g sin8s u+v

s=--t 2

The area under the v- 1 graph represents the di splacement. The slope of the v-t graph gives the instantaneous acce leration.

t Is

i ~

~ ~

l ~ L

.J ,~

: I

l

i

Chapter 2 Kine matics

Motion graphs of a bouncing ball (upward positive, with energy loss during impact)

T 5m

Termina l ve loc ity

G/ms' 1st bounce

q 1j H ll

0 1 -10 '!

V/ms'

2nd bounce

2 3

3rd bounce

tis

10 ~----"-------n-----,,

t Is

S/m 2 3 lis Oh.-----.----------------~

II l i li

-5

Air resis tance increases with speed of motion. Therefore, when an object fa lls in air, its acce leration falls gradually until zero . Then, the object mo ves with a constant speed, known as terminal velocity.

Motion graphs for a free-fallin g object in air (downward positive)

" -

"'

a v initial slope = g

constant velocity

s I t V'lope 4 ', IL1 . o,L-~------~--------~~

Brief Notes

27


WORKED EXAMPlES

EXAMPlE 1

SOlUTION

EXAMPlE 2

SOLUTION

EXAMPlE 3

SOLUTION

Fig.a shows a body's initial velocity v1 and final velocity v 2. Calculate its change in velocity .

..

1\/ /

EXAMPLE 4

SOLUTION

=

=

Chapter 2 Kinematics

The speed limit of the Tolo Harbour Highway is 100 km h- 1. (a) Express the speed limit in m s 1

A police radar speed check system detects a van speeding at its top speed of 40 m s1. Half a minute later, a police car, hiding at a distance 200 m ahead of the speed detector, accelerates from rest at a rate of 3.0 m s2 to its full speed of 54 m s1.

(b) How far must the police car travel before it reaches its top speed?

(c) Find the distance of the van ahead of the police car, when the police car has just reached its full speed.

(d) Calculate the velocity of the police car relative to the van wh en both cars are travelling at full speed s. Hence, calculate how long the police car takes to catch up with the van , after reaching its top speed.

Th d I. t v --100

x 103

-- 27.8 ms' e spee 1m1 IS 3600

(a)

(b) By v2 = u2 + 2a s, the d is tance covered by the police car is g ive n by 542 = 0 + 2. 3. s

(c)

(d)

.. s = 486 m

I -------~---~

speed detector police car accelerate.Jiwn rest

40 ms' van

=---c>

~---'------------~--=------------~~~-~~8--~-=~ l!f ~ ~ ----- 200 111 -~-----

Before the police car starts . the distance trave ll ed by the va n is s, = 40 x 30 = 1200 m

The time requ ired for the police car to reach its fu ll speed is v = u +at

t = 54

= 18 s 3

In this time interval. the van has travell ed s2 = -10 x 18 = 720 m a nd the po li ce car has travelled 486 m.

Since the police car is hidden 200 m ahead the speed c heck, the distance of the van ahead of the pol ice car is

d = 1200 + 720 - 486 - 200 = 1234 m = 1200 m

1234 Ill

The ve locity of the police car re lative to th e van is Vpv =up- Vv =54- 40 = 14 ms'1

1234 Thus, the time required is I=-~= 88 s .

14

Worked Examples

29

30 Section A Mechanic s

MU LTIPLE CHOICE QUESTIONS

Displacement, Velocity, Acceleration

1. A stone rolls along the curved path under the influence of the gravity.

2.

\Vhich of the followii1g statement is COJTect?

A . The horizontal speed of the stone is the same throughout the motion.

B. The vertical speed of the stone is the same throughout the motion .

C. The acceleration of the stone is the same throughout the motion.

D. The velocity of the stone is horizontal at each peak;.

A stone is ,released from rest. Which of the following graphs correctly represents the motion of the stone?

A.

B. s

0

C. "

s 0

Multiple-Choice Q u estions

3.

D. l' 1-------~-

0

A stone is thrown vertically upward. Which of the fo llowing quantities reverse(s) in direction at the highest position?

(I) velocity (2) displacement (3) acceleration

A. B . c. D.

(1) only (3) only (1) and (2) only (2) and (3) only

' 4. "'\ I A car moving at a speed of 10 m s suddenly

5.

6.

accelerates at 0.50 m s2 . What is the speed of the car after 5.0 s?

A. 7.5 m s 1 B. 10m s 1 C. 12.5 m s 1 D. 20m s 1

A coin is thrown vertically upward at 5.0 m s 1 from the top of a building. How long does it take for the coin to reach the ground which is 50 m below? Take g = 10m s2

A. 1.0 s B. 1.7 s c. 2.7 s b. 3.7 s

A train decelerates uniformly before it reaches a station which is 50 m ahead. If it takes 5 s to stop the train, what !stheillitial speed of the train?

(b. . 10m s1 B. 15ms1 C. 20m s 1 D. Cannot be determined because the

deceleration is not given.

I

f

Questions 7 to 8

7.

8.

For each of the questions below, choose one of the given velocity-time graphs which best represents the motion. Take the velocity upward as positive and neglect air resistance.

A. v :-~.

- -- ~ -~:-:- j ~---;----

B.

c. v

0

D.

A tennis ball is thrown up and returns to its original position.

A tennis ball is released so that it falls vertically onto the ground and bounces to its original position.

9.

L-


A stone takes _6.0 s_ to slide down a smooth inclined plane from rest. The plane makes an angle of 30 to the horizontal.

J

What is the length of the inclined plane? Take g = 10 ms2.

A. 30m B. 45 m 'C. 90 m D. 180m

10. Two cars, P and Q, are moving in the same direction with uniform speeds of 6.0 m s 1 and 10 m s 1 respectively. At 1 = 0 , P is 10 m ahead ~ ~ ofQ.

Q p '1---~ ~ .J

How long does it take for Q to catch up with p?

A. 2.0 s g:" ? -__ ) s c. 3.0 s D. 3.5 s

~-A ball is projected up a smooth inclined plane (/ which makes an angle of 60 with the horizontal.

It takes 4.0 s for the ball toreturn to the starting point. \~is the length of the inclined plane? Take g = 10 m s2

A. 8.7 m (B. 17.3 m c. 20.0 m D. 34.6 m


31

La f.., ..A>

32 Section A Mechanics ------------ -- -.

12. A bouncing ball is thrown vertically downward with an initial speed of 2 m s 1 from a height of I .5 m above the ground. Assuming no energy loss during the impact and neglecting air resistance, how long does it tike for the. ball to reach the original level again? ... Neglect air resistance and take g =10m s2 .

A. 0.38 s B. 0.77 s ! C. 1.00 s IV D. 1.50 s 4----.._, \

13. Two toy cars, A and B, movin at their(constant s eed towards each other, ~reach P - anaQ respectively at 1 = 0. Tney hit each other at X, such that PX : XQ = 2 : 3.1

---

p X Q

.:_;........, B

\Vhat is the ratio of the speeds of A to B?

A. 2 : 3 B. 3 : 2 c. 4 :9 D. 9:4 '

)

14. Two toy cars, A and B, are initially at rest at P and Q respectively. After released at the same , time, they ~ccelf

,

I . fy. A parachutist falls vertically with a uniform speed of 20 m s 1 At the moment when he is I 00 m from the ground, he drops a coin which accelerates. Neglect air resistanc Pill-and take g = ms2 What is the ti~ference between the coin and the parachutist reaching the ground?

A. 2.1 s B. 2.9 s c. 4.5 s D. 5.0 s

l

tone _P is !ocated a~a~ ther Jdent1ca! stone Q -Wh1ch 1s at a height h ~ Thet;o stones are released from rest at the same time. If the time interval between the stones hitting the ground is 2.0 s, what is the value of h?

A 80m B. 100m C. 115m D. 180m

19. A car travels up a hill at constant speed of 4 .0 m s 1 and returns down the hill at a constant ~of ~1 Calculate the average speed for the ro u n Clti'ip:"'

A. 2.4 m s 1

B. 4.8 m s 1 C. 5.0 m s 1

D. Cannot be determined because the distance up the hill is not given .

A car is travelling at a speed of 15 m s 1_and can be bwught to rest w'illi


25. Consider the velocity-time graph of an object. v

0

----------

Which of the followin g graphs best represents the coJTesponding variation of displacement s with time t?

A. s

B

C.

0

D. s


26. The graph shows the variation of velocity of an

27.

object with time. ,.

0

Which of the following statements is/are true?

(I) There is an instant when the object is at rest. V. There is an instant when the object changes its direction.

(2)

(3)

A. B. C. D.

There is an instant when the acceleration is zero. ~


A ball rolls down an inclined plane. '[he ball is firs t released fr~ and then later from Q.

Q

2h

0

Which of the following statements is correct ?

A .

B.

The ball takes twice as much time to roll X from Q to 0 as it does to roll from P to 0. The acceleration of the ball at Q is twice

f [

C.

D.

~-as large as the acceleration at P. ~ The speed of the ball at 0 is four times as "-,.!_ . much when rolling from Q as it is when . rolling from P. The ball takes Jess time to travel~ to 0 when rolling from Q than it does ~

when rolhng from P .

; i -

. f~;-~ . -

'

. . , ' ~-ailE.. , . ~-. l ~ !' t -~ 'i

~- I . .,:- 1 ~.~

~

.

*

28. Two identical balls, A and B, released from rest at the same time from P and Q respectively, slide down a smooth inclined plane as shown.

Q

p

21!

0 ~~~~~~ Which of the following statements is/are true?

(1) The speed of A when it reachx O is half that of B when Breaches 0. ,

(2) When A reaches 0, Breaches P. V (3) B takes a shorter time to travel from P to .

0 than does A .

A (1) only B. (3) only c (1) and (2) only D. (2) and (3) only

29. Two metal spheres P and Q are released from rest fro m a bridge 180 m above the river. .Q__is_ I:eleased 2 s after P What is the maximum vertical separation between P and Q in the air? Neglect air resistance and take g = 10 m s ~ .

A. 10m B. 90 m C lOOm D. 120m

30. A man at the top of a building releases a stone from rest. ~ater. he throws a marble downward with an initi~l_~locity . Ignore the effects of air resistanCe'.- whiC'l'iOnhe following graphs best represents the velocity-time graph for the stone (S) and the marble (M)?

A. ....

v s

r Is 0 2 3 4

ChapteT 2 Kinematics

B. v ;'v/ s

tIs 0

2 3 4

C. v M s

Is 0

2 3 4

B. \1 s M

0

3 I. At a construction site. workman A is in an elevator movi ng upward at a constant speed u. When the elevator is level with the first tloor. he ~ a spanner: At thesame instant, a second workman B standmg on the first floor throws a hammer upward with initial speed u .

Hammer thrown

~ II Ill

ElevatOr moving up

Which of the following statements correctly describes the motion of the hammer and the spanner as seen by the workman on the first

35

floor'l - '

Multiple-Choice Qu est io n s

36 l

Section A Mechanics I A. The spanner falls downward ahd the

hammer initially moves upward. I

B. The spanner lands on the~ at the same time as the hammer does.

C. The spanner lands on the floor before the hammer does.

D. The spanner and the hammer both move upward. But, the spanner rises more slowly.

Relative Motion and Motion in 20

32. A particle has an initia l. velocity of 15 m s1 in the Ox direction. At a later time, its velocity is 15 m s1 at an angle of 60 to Ox.

final velocity

initial velocity

0 X

What is the change of velocity that has taken place in this time interval?

A. 15 m s 1 at an angle of 30 to Ox. B. 15m s 1 at an angle of 120 to Ox. C. 26 m s 1 at an angle of 30 to Ox. D. 26m s 1 at an angle of 120 to Ox.

33 . A tennis ball travelling at 4 m s 1 due east strikes on a wall and bounces off at 3 m s 1 due north. What is the change in velocity of the tennis ball?

A. B. c. D.

1 -1 ms 5 m s 1 5 m s1 5 m s1

due south S 53W N53W N37W


..

34. A boat has a maximum speed of 4.0 km h-1 in still water. It is moving across a river where the speed of flow is 2.0 km h-1 If th ~ ca~ain wishes to ~th s.honest mute (perpendicular to the stream flci~) a!.Jh.e..maxim~ eed , how should he s.teer the Q.oat upstream?

A. 30.0 with the stream B. 45.0 with the stream C. 60.0 with the stream D. 63.4 with the stream

35. A moving belt of width 3.0 m is running at a constant speed of 2 m s1 to the right. A cat can walk with a maximum speed of 1.5 m s 1 on the ground. It walks across the belt from P using the shortest ti~

movinc belt a1 2 m ~- 1 to the right

~

Find the time needed and the position of the cat when it reaches the far side.

Shortest time Distance from R

A. 2s 0 B. 2s 4m C. 6s 3m D. 6s 4m

..

' +

36. Two bodies X and Y are moving with constant velocities in the direction indicated by the arrows. At time t == 0, they are at the position shown.

37.

c..-x

... .......... . r-{_; t 3m s 1 y II

8m

At time t == 2 s, the magnitude of the velocity of Y relative to X is

A. 4ms' B . 5 ms' C. 6 m s' D. 7 ms'

A man is walking due east at 1.00 m s' on the deck of a ship steaming due north at 1.73 ms' . In what direction will the man be walking relative to the surface of the earth?

A. N 30E B. N 60 E C. NE D. N 40.9 E

I


38. A boy wished to swim across a river with parallel banks as shown. Assume that the water current is flowing at 0.2 m s', and the swimming speed of the boy is 0.4 m s'.

39 .

water current =---c>

Along which direction shou ld the boy swim if he wishes

(1) to take the shortest route to the opposite bank?

(2) to reach the opposite bank in the shortest time?

A. B. c. D.

( 1) shortest route R Q Q R

(2) shortest time Q R p p

When a man is running due west, he feels that the wind is blowing towards him from the north . What is the actual direction of the wind?

A. from the east B. from north-east C. from south -east D. from south-west

40. If two ve locit ies each of magnitude v and one of which is directed due north a~e tompounaeO, which of the following statements concerning the resultant velocity is/are true?

(1) (2) (3)

A. B. c. D.

It may be zero. It may act due north. It may act due east.



37

38 Section A ~: ._.

Mechcmics

EXERCISES

In all calculations, take g =10m s2 and ignore air resistance unless specified otherwise.

VElOCITY , RElATIVE MOTION

I. Calculate the total displacement in each of the following cases:

(a) a boy who walks 20 m south and then 20v2 m south-east, (b) a girl who runs halfway round a circular 400 m running track, starting at its most easterly point, (c) a boat moving with top speed upstream for 10 s and then downstream for another 10 s, the top

speed of the boat being 2.5 m s 1 in sti ll water and the river current being 2.0 m s-1 downstream.

2. A light pulse lasts only I0-14 s. What is the length of the pul se? Take the speed of light in air to be 3.0 x 108 m sl.

3. In an athletic meet, the time for a runner on a 100m track is recorded by a stop-watch to be 12.2 s. lf the largest probable error of the stop-wa tch is 0.20 s and th at of the track length i s 0.5 m, \\hat is the largest probable error of the average speed of the runner? Do you think the given uncertainty in track length acce ptable ?

4. A car and a truck , moving at steady speeds of 30 111 s 1 and 25 111 s 1 re spectively, pass a lamp-post on a straight motor way at the same time. Calculate their distance of separation after 3.0 minutes, if they are traveling

(a) in the same direction as shown , (b) in the opposite direction.

5. A boat travels downstream along a river at a speed of 8.0 m s 1 and returns moving against the current at 4.0 m s 1. What is the average speed of the boat?

6. A shell is fired from a gun with a speed of 20 m s- 1 relative to the gun, which recoils with a speed of 0.20 m s 1 relative to the ground. Determine

(a) the recoil speed of the gun relative to the moving shell, and (b) the speed of the shell relative to the gro und.

7. A lorry is moving in a straight line at a constant speed of l 0 m s 1 A package is thrown from the lorry with an initial horizontal velocity of 2.0 m s1 relative to the lorry , and in a direction opposite to its motion. The package stays in air for 0.50 s.

(a) What is the initial speed of the package relative to the grou nd ? (b) What is the horizontal separation between the lorry and the package when the package hits the

ground?

Exercises

,.~ . ..

Chapter2 Kinematics

8. An aircraft from U.S.A. is scheduled to reach the H.K. International Airport at 6:30p.m. However, it is speeded up by a steady wind so that it arrives H.K. earlier at 5:40 p.m. If the distance of U.S.A. from H.K. is 7200 km and the average speed of the aircraft relative to the earth surface below is 100 m s- 1, calculate

(a) the departure time at U.S.A. according to the clock in H.K., (b) the actual speed of the aircraft (speed relative to air), and (c) the average speed of the wind.

[Hint: Since the aircraft has arrived earlier than scheduled, the wind is blowing in the direction of travel. As a result, the actual speed of the aircraft is less than 100m s- 1. ]

9. Calculate the change in velocity in each of the following cases:

(a) a boy who walks at steady speed of 0.20 m s- 1 due south and then at steady speed of 0.2v'2 m due south-east,

(b) a girl who runs with steady speed of 8.0 m s- 1 halfway round a circul ar running track, starting at its most easterly point due north,

(c) a boat moving with top speed of 2.5 m s- 1 upstream and then downstream, the river current being 2.0 m s- 1 downstream.

* 10. On a rainy day, the path of each raindrop makes an ang le of 20 with the vertical as a wind blows in the northerly direction. A passenger in a car travelling due north at 15m s- 1 sees perfectly vertical tracks of rain on the windowpane. Determine the speed of the raindrops relative to the earth .

* 11. A plane travels with a speed of 100m s- 1 in still air . A 10m s-1 ,,ind is blowing. The resultant speed of the plane is still 100m s- 1 and the resultant path is in the northerly direction. What is the direction of the wind?

* 12. A plane can attain a speed of 110m s- 1 when there is no wind. The pilot sets out for a destination 700 km due north.

(a) The pilot find s that there is a wind blowing in the easterly direction (relative to the earth). As a result, the plane must be headed 25 west of north in order to fly there directly. How long does it take for the plane to arrive the destination?

(b) A wind is blowing but the exact direction is unknown. To fly the destination directly, the plane must be headed 25 west of north and the total time of flight is 3.0 h. What are the magnitude and direction of the wind velocity?

ACCELERAT I ON

13. A train slows down smoothly from a speed of 100 km h- 1 to rest within a distance of 1.0 km. Estimate its average deceleration.

14. When brakes are applied, a car can be brought to rest from a speed of 20m s- 1 in a time of 1.5 s. If the driver's reaction time is 0.20 s, calculate the total braking distance of the car moving at an initial speed of 30m s- 1.

15 . The minimum speed required for a plane to take off is 120m s- 1 If the acceleration of the plane on the ground is 30m s-2 , what is the minimum length of runway?

ii Exer cises 0

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Section A Mechanics

16. A man stands on the edge of a cliff and throws a stone over the edge vertically upwards at 15 m sl. After what time will the stone hit the ground 50 m below?

17. A stone is projected up along the greatest slope of a smooth inclined plane with an initial speed of 1.5 m s 1. If the acceleration of the stone is always 2.0 m s2 down the incline. calculate

(a) the di stance travelled by the stone up the inclined plane, (b) the time taken for the stone to return its startin g point , and (c) the average speed up and down the plane.

18. A ball is released from rest at a height of 45 m abo ve the ground. Another ball is then released exactly 1.0 s after the first ball is released. When the first ball reaches the ground, calculate

(a) the speed of the second ball, and (b) the height of the second ball above the ground.

19. A half-metre rule is initially held vertically with the zero mark at the upper end. Timothy is ready to catch the rule between hi s thumb and first finger at the lower end. The rul e is rel eased sudd enly and Timothy catches the rule at its 30 em mark. Calculate the reaction time of Timothy.

20. A water-rocket accelerates from rest at ground leve l at 5.0 m s 2 vertically upwards for 30 s and then it undergoes free-falling in air. Calculate

(a) the speed and the height of the rocket at 30 s after laum:h, (b) the maximum height reached by the rocket, (c) the time for which the rocket is air-borne, and (d) the speed of the rocket just before it hits the ground.

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21. A MTR train accelerates from rest at one station at a rate of 1.5 m s 2 for half of the distance to the next station, then decelerates for the final half. If the time of acceleration equals the time of deceleration and the stations are 1500 m apart, find

(a) the maximum speed of the train, (b) the deceleration of the train, and (c) the total time of travel between the two stations. (d) Draw a velocity-time graph to represent the motion of the train from one station to the next.

22. When the traffic light turns green, a car accelerates from rest at a rate of 2.0 m s2 . At the same time, a truck, moving at a constant speed of 10m s 1, passes the traffic light, which just turns green , and overtakes the car. Since the car moves with increasi ng speed , it will overtake the truck some times later.

(a) At what distance from the traffic light will the car overtake the truck? (b) What is the speed of the car at that instant?

23. The maximum acceleration (and deceleration) th at is tolerable for passengers in a KCR train is 1.5 m s 2 and the speed limit of the train is 40 m s l.

(a) What are th e minimum accelerating distance and the ll111111llUm accelerating time of a KCR train. assuming that it will move at full speecl7

(b) If two sta ti ons are 6.0 km apart , what is th e minimum time of trave l bet ween these two stations"

Exercises

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24.


An ambulance is speeding along the fast-lane of a highway. As there is some road works in the fast-lane, it has to cut into the slow-lane. As soon as it shifts into the slow-lane, the driver is shocked to find a truck 100 m ahead moving at a slow steady speed of 0.50 m s 1. At that instant , the speed of the ambulance is 40 m s 1 and the driver of the ambulance immediately applies the brake . What must be the deceleration of the ambulance if a collision is to be just avoided?

25. A ball is dropped from rest at a certain height hand strikes the ground with a speed of 20m s 1

26.

27 .

28.

(a) Find the value of h . (b) How long does it take to fall

i) the first h/2, ii ) the second h/2?

An object falls from a bridge that is 55 m above the water. It falls directly into a small boat moving with co nstant ve locity v. If the boat was 14 m from the point of impact when the object was released, what was the speed of the boat?

A shell is shot vertically downward from a gun. The speed of the shell just before it strikes the grou nd is 250m s 1. If the she ll imbeds itself 0.40 minto the ground, find

(a) the average deceleration requi red to stop the shell, and (b) the time required fo r it to come to rest.

At the Science Museum, balls are released one by o ne from a machine at a regular time interval of 12 s. The ball ro ll s down a track with an average acceleration of 0.2 m s 2 When the first ball reaches the end of the track, the lOth ball begins to fall. Determine the di stance of separation betwee n

(a) the firs t and the second balls, (b) the 9th and the lOth balls ?

29. A ball is dropped from the Phys ics Lab of a sc hool building. One second after the first ball is dropped, another ball is thrown vertically down ward with an initial speed of I 5 m s 1 The two balls strike the playground at the same time.

(a) Determine the height of the Physics Lab above the playground. (b) Plot a speed-time graph for each ball, taking the instant the first ball was released to bet= 0.

30. A balloon is ascending at the rate of 16 m s 1. At a height of 120 m above the ground , a package IS dropped from rest relati ve to the balloon. How long does it take for the package to reach the ground?

31. An open elevator is ascending with a constant speed of I 0 m s 1 A ball is thrown straight up by a ma n on the elevator when it is at a heigh t of 25 m above the gro und. The initial speed of the ball relati ve to the elevator is 20 m s 1 (a) What is the maximum he ight attained by the ball above the ground? (b) How long does it take for the ball to return to the elevator?

Exercises

41


EXAM-TYPE QUESTIONS

1. vlms'

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The velocity-time graph shows the first 1.6 s of the motion of a ball which is thrown downward at 6 m s I The ball hits the ground and rebounds directly upwards. taking a negligible time to bounce.

(a) How far does the ball travel before first hitting the ground? (2 marks) (b) Calculate the change in velocity as the ball rebounds. (I mark) (c) How high does the ball rise after rebounding? (2 marks) (d) Calculate the acceleration of the ball between C and D. (1 mark) (e) At what time does the ball next reach the ground? (1 mark)

(f) If the ratio of the downward velocity .to the upward velocity has a constant value, determine the upward velocity of the ball on its

i) second rebound,

ii) third rebound. (2 marks)

(g) Extend the velocity-time graph to illustrate the motion of the ball up to the highest position after the third rebound. (3 marks)

Exam-Type Questions

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2 . (a) Determine whether each of the following situations is possible or not . lf it is possible, give an

(b)

example to support your answer.

i) A body instantaneously at rest is accelerating.

ii) A body has a constant velocity but varying speed.

iii) A body has varying velocity but instantaneous zero acceleration. (3 marks)

An astronaut has reached a planet where the atmosphere is absent and the acceleration due to gravity is l m s2 .

i) A ball is projected vertically upward with a speed of 2 m s 1 When the ball returns, it is caught at a point 2m vertically below where it is projected. The motion of the ball is recorded in the acceleration-time (a-t) graph which is partially drawn in Fig.a. Downward is taken as positive.

i i)

0 L---------------------------L---~

Fig.a

( l) Complete the a-t graph in Fig.a by putting appropriate numbers on the time-scale. Also. mark on the graph to show when the ball was at its highest position.

(2) By taking downward as positive, carefully sketch the velocity-time (v-1) graph and displacement-time (s-1) graph of the ball. (4 marks)

A gun is placed with its mouth just below the surface of a long column of viscous liquid as shown in Fig. b. A bullet is fired from the gun. It is noticed that after a while, the bullet falls with a uniform speed of 2 m s 1 downward. The a-t graph which represents the motion of the bullet is recorded in Fig.c. Again, downward is taken as positive. The e~periment is carried out on the planet.

a /ms' 10 I Is

Fig.b Fig.c

( l) Estimate the initial firing speed of the bullet.

(2) Roughly sketch the v-t graph of the bullet.

(3) If the gun is placed at the bottom of the liquid column and the bullet is fired vertically upward, estimate the initial acceleration of the bullet. (5 marks)

Exam-Type Questions

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Section A Mechanics

3. A motion sensor used in a laboratory sends out short pulses of ultrasound at a regular time interval of 0 .05 s. Given that the duration of each pulse is 1 ms and th e speed of sound in a ir is 340 m s 1

(a) There are tw o limitations of the device:

I. Whil e the de vice is transmitting a pul se, it cannot detect any refl ecte d pul ses. 2. The echo of th e first pul se mu st arri ve before the second pulse is transmitted; Otherwise, th e

f irst pul se will be ignored .

i) What is th e mi nimum range (di stance of th e objec t to be measured) of th e dev ice'i ii ) What is the max imum range of th e dev ice? (4 marks)

(b) In measuring th e speed of a tro lley m ovin g at a constant ve loc ity . th e ro und-tri p t ime of the first and the second pul ses are measured to be 8.1 ms and 9.4 ms res pecti vely.

i) Find th e pos ition of th e trolley from th e dev ice when (1 ) the first pul se is refl ected (2) th e second pul se is refl ected

ii) Hence, fi nd the average speed of th e tro ll ey.

iii ) Di sc uss whether th e ave rage speed in ii) is eq ual to th e in stanta neous speed of the tro ll ey or not. ( 4 ma rks)

(c) A tro ll ey is pl aced on an inclined pl ane whi ch has a sprin g fi xed near th e botto m as shown in Fig.a . As soon as th e tro ll ey is re leased , th e moti on sensor starts to reco rd th e speed of the tro ll ey. A velocity- time grap h is pl o tted as shown in Fi g. b.

i)

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moti on sensor

Fig. a

Find the slopes of th e v- 1 graph when the troll ey is

(l ) movin g down th e plane and (2) mov ing up th e plane .

1 Is

ii ) Estim ate th e angle of in clin ati on of th e inclined pl ane, statin g any ass ump tio n(s) yo u have made. (4 marks)

Exam-Type Questions

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mechanics exercise

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