Oscillations 2008 Prelim Questions

1. A particle moves in simple harmonic motion according to x = 2 cos(50t), where x is in meters and t is in

seconds. Its maximum velocity in m s-1

is:

A 100 sin(50t) B 100

C 200 D 5000

ACJC 08 H2 P1 Q14

2. A particle in simple harmonic motion at a particular point in its motion has a kinetic energy of 5 J and a

potential energy of 3 J. Given that its potential energy at its equilibrium point is 0 J, which of the

following statement is correct?

A At its amplitude, the kinetic energy is 8 J and the potential energy is 0 J.

B The average kinetic energy is greater than the average potential energy.

C The average kinetic energy = the average potential energy = 4 J.

D Its potential energy varies from – 8 J to + 8 J.

ACJC 08 H2 P1 Q15

3. An oscillator is subjected to a damping force that is proportional to its velocity. A periodic driving force is

applied to it. Which of the following statement is correct?

A Its amplitude increases continuously with time.

B Its amplitude decreases continuously with time.

C Its amplitude would remain constant after some time.

D Its amplitude initially increases then decreases.

ACJC 08 H2 P1 Q16

4.

a. A rectangular block of wood of cross-section A and thickness t floats horizontally in a water as

shown in Figure 5.

The block floats when its lower face is at a depth d in the water of density ρ. The block experiences a

force F on its lower surface as a result of immersion in the water.

i) State the direction of the force F.

The direction of the force F is [1]

ii) The pressure on the lower surface of the block due to the water is P. Show that P is

related to d, ρ and the acceleration of free fall g by the expression

P = dρg [2]

iii) Using the expression in (ii), show that the force F is related to the volume V of water

displaced by the expression

F = Vρg [2]

b. When the block is pushed down a further distance x into the water, show that the expression of the

resultant force FR is given by

FR = Axρg [2]

c. When the block is pushed downwards and then released, it undergoes damped simple harmonic

motion.

i. Using the result from (b), explain why the block is said to be undergoing simple harmonic

oscillation [1]

ii. Hence state the expression of the angular frequency of the motion of the block in terms of A, ρ,

g and m where m is the mass of the block.

Expression for Angular frequency = [1]

iii. Explain why the simple harmonic motion of the block is damped. [2]

iv. Sketch a labeled graph showing the variation of displacement x with time t for a time interval

over three periods.

[2]

d. Surface water waves from a constant amplitude wave generator are incident on the block. These

causes forced oscillations in the motion of the block. The frequency of the wave generator is

varied and resonance was observed at a particular frequency.

i. Explain the terms given in italics with reference to the motion of the vibration of the block.

Forced oscillations:

Resonance: [4]

ii. Resonance occurs when the water waves incident on the block has a speed of 0.90 m s−1 and wavelength 0.30 m. 1. Calculate the frequency of the water waves.

Frequency = Hz [1]

2. Given that the expression for the natural frequency of the oscillating block is given by

where f is in Hz and m in kg, calculate the mass of the block.

Mass = kg [1]

3. Describe and explain what happens to the amplitude of vertical oscillations of the block after the block has absorbed some water. [1]

ACJC H2 P3 Section B Q6

5. A horizontal plate is vibrating vertically with simple harmonic motion at a frequency of 20 Hz. What is the

maximum amplitude of vibration so that the mass on the plate always remains in contact with it?

A 6.2 x 10-4

m B 7.8 x 10-2

m

C 5.0 x 10-2

m D 4.9 x 10-1

m

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6. Two identical pendulums P and Q are given the same initial displacement and then released. P and Q

are then subjected to oscillatory driving forces of the same magnitude and of variable frequency f.

If P oscillates in air and Q oscillates in water, which one of the following graphs represents the variation

with f of the amplitudes of P and Q?

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7. Part (a) – (c) of this question focuses on circular motion.

d. Some parents rock their babies to sleep by using a cradle attached to the end of a fixed spring

as shown in Fig. 5.2 below.

Fig 5.2

A 5.0 kg baby is placed in a cradle of negligible mass and given a downward displacement of 20.0 cm

such that it undergoes vertical simple harmonic motion.

i. Show that the angular frequency of oscillation ω is given by the equation

where m is the mass of the baby and k the force constant of the spring. [2]

ii. Given that k = 20.0 N m-1

, calculate for the system of cradle and baby

1. the period of the oscillation,

period = s[1]

2. the kinetic energy as it passes through the equilibrium position,

kinetic energy = J [1]

3. the maximum net force experienced.

Maximum net force = N [1]

iii. Sketch a labelled graph of the kinetic energy of the system against displacement. [2]

iv. The baby’s father wants to take a picture of his child oscillating in the cradle. Unknown to him, his

camera exhibits a shutter delay of 3 s. If he depresses the button when the cradle is at the highest

point, how far is the cradle away from its expected position in the picture?

Distance from expected position = m [2]

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8. Which of the followings is incorrect between the two oscillations X and Y as shown in figure below?

A At time t = 0, X leads Y by π/2 radian

B The phase difference is constant

C The amplitude of X is twice that of Y

D The total energy of X is the same as that of Y

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9. A particle is oscillating with simple harmonic motion described by the equation:

y / m = 5 sin (20π t/s)

How long does it take the particle to travel from its position of maximum displacement half way to the

equilibrium position?

A Less than 1/80 s

B 1/80 s

C Between 1/80 s and 1/40 s

D 1/40 s

CJC H2 P1 Q15

10. A trolley is resting on a frictionless and horizontal surface. Initially, the two identical springs are at their

natural lengths. A thin wooden board is attached vertically to the top of the trolley as shown in Fig. 1.1.

The total mass of the trolley and the board is 5.00 kg.

a. The trolley is displaced to the right by 10.0 cm. At the instant when it is released, a stop-watch is

activated. The period of oscillations is found to be 2.00 s.

i. Calculate the initial potential energy of the system, given that the equivalent spring constant of the

‘combined’ spring is 1.00 kN m-1

.

Initial potential energy = _____________ J [1]

ii. Taking into account the effect of air resistance, sketch, on Fig. 1.2, a well- labelled graph of

potential energy against time for this system for the first 4.0 s.

b. The same trolley is again displaced to the right by 10.0 cm and released. An air jet mounted on a wall P,

as shown in Fig. 1.3 emits continuous ‘pulses’ of air at a frequency of 0.500 Hz.

When the trolley is at its equilibrium position and moving towards the left, a ‘pulse’ of air from the air jet

arrives at the centre of the wooden board. Describe the variation of the amplitude of the subsequent motion

of the trolley. [2]

c. Question changes into a circular motion question of a different context.

HCI 08 H2 P3 Section A Q1

11. The bob of a simple pendulum moves through its equilibrium position at time t = 0 s. Which of the

following graphs best shows the variation of the acceleration a of the bob with time t?

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12.

12.

a. During a basketball training session, two players were tossing the basketball to and fro to each other

repeatedly. Explain why the motion of the basketball is not a simple harmonic motion. [1]

b. An object mass m is attached to a horizontal spring as shown in Fig. 3.1. It is then pulled to the right by

a distance x0 from its equilibrium length before it is released. It subsequently exhibited simple harmonic

motion.

i. When the object is at a distance x from its equilibrium position (x < xo), it has a speed v. The spring

constant is given to be k. Based on conservation of energy, write an equation that relates m, k, v, x

and x0. [1]

ii. It is given that velocity of a particle in s.h.m. is . Use the equation you obtained in b

(i) to prove that . [1]

iii. It can be further proven that from . However, when experimentalist plots T2

versus m, the graph does not pass through the origin. Suggest one possible reason for the source

of this systematic error. [1]

c. (circular motion)

An object is whirled in a vertical circle. In Fig. 3.2, v represents the velocity of the object, vx is the horizontal

component of the velocity, R is the radius of the circle and x is the projection of the object on the horizontal axis.

By obtaining two expressions for sin q, show that vx is of the form .

[3]

IJC 08 H2 P2 Q3

13. A particle performs simple harmonic motion according to the equation x = 2 cos ωt where x is measured

in cm and t in s.

If the angular frequency ω is π rad s-1

, what is the total distance travelled by the particle at t = 1.5 s?

A 0.0 cm B 2.0 cm C 3.0 cm D 6.0 cm

JJC 08 H2 P1 Q21

14. Because of air resistance, the amplitude of oscillation of a simple pendulum decays exponentially with

time. How does the total energy of the pendulum vary with time?

A It remains constant.

B It decays at a steady rate.

C It oscillates about zero with the same frequency as the pendulum.

D It decays exponentially.

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15.

a. State the equation defining simple harmonic motion. [1]

b. A student clamps one end of a flexible plastic ruler against the laboratory bench and flicks the other end,

setting the ruler into simple harmonic oscillation. The end of the ruler moves a total distance of 8.0 cm

as shown in Fig. 4.1 and makes 28 complete oscillations in 10 s.

Fig 4.1

i. Find the angular frequency ω of the oscillation. [1]

ii. Fig. 4.2 shows the variation with time t of the displacement x for this oscillation.

Write the equation for this oscillation. [1]

iii. What is the maximum speed of the plastic ruler? [2]

iv. Sketch a graph of velocity v against displacement x for this motion. [2]

v. The end of the ruler is attached with a piece of card of large surface area and the experiment is

then repeated. Sketch a graph on Fig. 4.2 show the effect of this change on the variation with t of

the displacement of the ruler. [2]

JJC 08 H2 P2 Q4

16. A system oscillates under the influence of an external periodic driving force. Which of the following

statements is incorrect?

A In steady state the system vibrates at the frequency of the driving force.

B The amplitude of vibration becomes very large when the frequency of the driving force is close

to the natural frequency of vibration of the system.

C The amplitude of vibration remains finite if damping forces are present.

D At resonance the displacement of the system is in phase with the driving force.

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17. Which is not an example of approximate simple harmonic motion?

A A ball bouncing on the floor.

B A child swinging on a swing.

C A piano string that has been struck.

D A car’s radio antenna as it waves back and forth.

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18. A small delivery truck can be thought of as a box supported by four springs, one at each wheel (the

suspension of the truck).

On a particular road, speed bumps are put on the road to slow down the traffic. After passing rapidly

over one of these speed bumps, a delivery truck experiences rapid vertical oscillations.

Figure 3.1

Figure 3.2 shows a graph of acceleration, a, against displacement (from equilibrium), x, for the motion of the

truck.

a. Calculate the angular velocity ω of the truck.

ω = ………………… rad s-1

[2]

b. Calculate the shortest time taken t for the truck to oscillate from its lowest point to a point 0.025 m below

its equilibrium position.

t = ………………….. s [3]

c. If the truck travels at a certain speed over the series of speed bumps, the vertical oscillations can be

very large. Explain why this is so. [2]

MJC 08 H2 P2 Q3

19. A particle of a mass of 90.0 g undergoes simple harmonic motion. The graph in Fig. 16 shows the

variation of its kinetic energy, EK with time, t.

What is the maximum acceleration of the particle?

A 0.074 m s-2

B 0.148 m s-2

C 37 m s-2

D 74 m s-2

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20. A loaded test-tube floats upright in a liquid of density ρ, with a length submerged as shown in Fig. 2.1

below. You may assume the cross-sectional area of the test-tube A is constant.

a. Write down an expression for the upthrust acting on the test-tube by the liquid. [1]

b. By considering the resultant force on the test-tube when it is displaced downwards by a small distance x

from the equilibrium position,

i. draw the free body diagram of the test- tube at the instant of displacement x; [1]

ii. show that the subsequent motion upon release is simple harmonic; [3]

iii. hence show that the period T of the simple harmonic motion is given by

2l

Tg

[2]

NJC 08 H2 P2 Q2

21. The displacement x of a body of mass 0.02 kg in simple harmonic motion is given by the equation

x = 5.0 x 10-3

sin (6 π t),

where x is in metres and t in seconds. Which of the following is true about the maximum velocity and the

maximum restoring force?

Maximum velocity Maximum restoring force

A 1.5 x 10-3

m s-1

9.0 x 10-4

N

B 1.5 x 10-2

m s-1

9.0 x 10-5

N

C 3.0 x 10-2

m s-1

3.6 x 10-3

N

D 9.4 x 10-2

m s-1

3.6 x 10-2

N

NYJC 08 H2 P1 Q18

Fig. 2.1

22. The figure below shows two simple pendulum X and Y having the same lengths and same bob sizes

and a simple pendulum Z whose length can be varied.

The pendulum X and pendulum Y will oscillate after the pendulum Z is set into oscillations in a plane

parallel to the wall. Which of the graphs below shows the variation of the amplitude a for the pendulum X

and pendulum Y with frequency f of the pendulum Z when the length of pendulum Z is varied?

A B C D NYJC 08 H2 P1 Q21

23. A trolley of mass m with free running wheels is attached to two fixed points P and Q by two identical

springs of force constant k. The trolley is then displaced small distance x towards Q and then released.

The acceleration of the system is given by the expression a = -10 x.

What is the period of the oscillation?

A B

C D

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24. A particle of mass 0.20 kg moves with simple harmonic motion of amplitude 0.050 m. If the total energy

of the particle is 4.0 x 10-3 J, then its period of motion is

A s B s C s D s

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25. Two identical bar magnets P and Q are suspended from identical helical springs so that one pole of the

magnets lies within identical coils of wire as shown. The resistance of the resistor RP is larger than that

of resistor RQ.

P and Q are then subjected to driving forces of the same constant amplitude and of variable frequency f.

Which graph represents the variation with f of the amplitude A of P and Q?

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26.

a. Describe what is meant by simple harmonic motion. [2]

b. Describe how, for a simple harmonic motion, the direction of acceleration varies with the direction of the

velocity. [2]

c. A long pendulum performs small oscillations and the position of the bob at 0.10 s intervals is shown in

Fig. 2.

Fig. 2 shows exactly one half-cycle of its motion. If the motion is simple harmonic, what is the maximum

speed vmax of the bob?

vmax = m s-1

[3]

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27. Fig. 13.1 below shows the variation of the kinetic energy with time of a particle undergoing simple

harmonic motion with amplitude of 0.30 cm.

What is the maximum acceleration of the particle in m s-2

?

A 0.094 C 1.8

B 0.74 D 3.0

SAJC 08 H2 P1 Q13

28. A vertical spring supports a mass M of 0.8 kg, as shown in Fig. 1.1 below. The mass is displaced

upwards by 2.0 cm and then released. It performs simple harmonic motion. The frequency of oscillation

is 2.0 Hz.

a. Define simple harmonic motion. [1]

b. At the point the mass M is released, calculate the net force acting on the mass. State the direction of

this force.

net force = …………………N

direction = ………………… [ 3 ]

c. At the point the mass M is released, calculate the net force acting on the mass. State the direction of

this force.

kinetic energy = …………………J [ 2 ]

d. State the energy conversions when the mass moves from the equilibrium position to the lowest point. [2]

SAJC 08 H2 P3 Q1

29. This question is about the oscillation of a mass between a pair of springs as shown in Fig. 4.1.

Fig. 4.1 a. The system obeys Hooke’s Law with a stiffness constant k. The block is displaced a horizontal distance

x and released.

i. Show that the initial acceleration a of the mass m is given by [2]

ii. Explain why the equation in (i) shows that the body will undergo simple harmonic motion. [2]

b. Such a system is used as a damper to reduce the movement of tall buildings in earthquakes or high

winds as shown in Fig. 4.2.

Fig 4.2

The system is designed to reduce the oscillations of a building which has a natural frequency of 0.50 Hz. A sudden movement of the building displaces the block 0.70 m from its equilibrium position relative to the building.

If the stiffness constant k of the system is 2.8 x 10

6 N m

-1, find the energy transferred to the oscillator.

energy transferred = J [1]

c. The oscillator is damped. It loses 50% of its energy on each oscillation. Find the amplitude of the

oscillator after one complete oscillation.

amplitude = m [2]

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30. A mass of 0.40 kg undergoing simple harmonic motion has a total energy of 5.0 J and amplitude 0.30 m. What is its period?

A 0.023 s B 0.38s C 2.7 s D 17 s

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31. A pendulum with a period of 2.0 s starts its swing from its extreme position. How long does it take to reach halfway between its extreme position and equilibrium position?

A 0.17 s B 0.25 s C 0.33 s D 1.0 s

VJC 08 H2 P1 Q16

32. Which of the following graphs best shows how the velocity v of an object undergoing simple

harmonic motion with amplitude xo varies with displacement x? The arrow on each graph gives

the direction in which the graph would be mapped out as time progresses.

YJC 08 H2 P1 Q14

33.

a. For a particle undergoing simple harmonic motion (SHM), write down the equation that shows

the relationship between its acceleration a, frequency f and displacement x. [2]

b. The speed c of an electromagnetic (EM) wave is given by a simple ratio as shown below

whereby E and B are the magnitudes of the electric-field vector and magnetic-field vector in the

wave.

Determine the amplitude of E in an EM wave when the maximum value of B is 1.00 pT.

amplitude of E = …………… V m−1

[1]

c. When the EM wave in (b) having a frequency of 200 kHz passes through a region of free

electrons, these electrons oscillate with SHM. Determine the amplitude of oscillation of the electrons.

amplitude = …………… m [4]

d. For an electron in (c), determine the minimum uncertainty in its velocity.

Minimum uncertainty in velocity = …………… m s−1

[2]

YJC 08 H2 P2 Q5

34. The graph in Fig. 17 shows how the displacement s of a body varies with time t when it is oscillating with

simple harmonic motion.

Which one of the following statements about the body is false?

A The kinetic energy of the body is a maximum at Q.

B The kinetic energy of the body at P is the same as the potential energy of the body at Q.

C The body has equal amount of kinetic energy and potential energy at R.

D The body has minimum kinetic energy at P.

SRJC 08 H2 P1 Q17

35. Which one of the following could be an effect of critical damping?

A A toilet door outside LT5 takes a long time to close after a student enters.

B A passenger in a car hardly notices that the car has just crossed a hump.

C A rubber ball drops to ground and bounces to its original position.

D A voltmeter fluctuates several times before registering a steady reading.

SRJC 08 H2 P1 Q18

36. A block of wood of mass m floats in still water, as shown in Fig. 3.1.

The block is pushed vertically into the water by 1.25 cm, without totally submerging it. It is then released

with an initial acceleration of 4.50 m s−2

. As a result, it bobs up and down in the water in simple

harmonic motion.

a. Calculate the frequency of the bobbing motion

b. Calculate the displacement of the block from its initial position after 0.020 s.

c.

i. Surface water waves of frequency increasing from zero are then incident on the block. At a

particular frequency, the block was observed to move up and down with an observable increase in

amplitude. Explain the cause of the observable increase in amplitude.

ii. State what happens when the block has absorbed water after some time.

d.

i. Sketch a graph to show how the amplitude of the oscillation of the block changes with the frequency

of the wave, showing appropriate values where possible.

ii. On the same axes, sketch the graph to show how the amplitude of the oscillation would be, if a

flatter block of the same mass and volume, as shown in Fig 3.2 was to be oscillating instead.

SRJC 08 H2 P3 Q3