Notes to the student:

Don’t leave out the qualitative questions, that is the ones with no numbers, just explanations. They are real physics too !

If a question or section is labelled “ENR”, then your teacher will tell you whether you are responsible for this kind of question.

A question labelled “qual” is qualitative, ie has no numbers.

A. Rotational Kinematics

A01 [ easy] An object is turning clockwise at a constant 150 radians/sec. a) Find its average angular acceleration. b) How long does it take to do 400 full rotations?

A02 [easy]

An object is initially spinning at 4 rotations/sec clockwise. After 0.2 sec it is turning CCW at 500° /sec. Assume that the angular acceleration is constant. a) What was its average angular acceleration in radians/s2? In degrees/sec2? b) Find its angular velocity when it has rotated 2 radians CCW from its initial orientation. c) When, if ever, did it stop for an instant?

A03 [easy]

Assume that the object in ex 1 is rigid, and consider two points on the object. Point P1 is 2 cm from the axis of rotation. At t = 0.5 s, point P2 is moving at 72 cm/s. a) At t = 0.5 s, what is the angular velocity, what is P2’s linear speed, and how far is P2 from the axis? b) Find the magnitudes of radial and tangential acceleration of P1 at t = 0.5 sec.

A04 [easy]

A certain wall clock has three hands: The hour hand (which goes around once every 12 hours) is 5.0 cm long. The minute hand is 8.0 cm long, and the second hand 12.0 cm. a) For each hand, find its angular speed, and the linear speed of its tip. b) What is the linear speed of a point on the minute hand, 2.0 cm from the tip?

A05 [easy] [extra practice] An object is rotating slowly counterclockwise. At t = 1.0 sec, a radial line on the object is pointing 20o South of East. At t = 5.0 sec, it is pointing 40o N of E. Find the average angular velocity, in o/s and in rad/s.

A06 [easy] [extra practice]

A diver makes 2.5 revolutions on the way from a 10-m-high platform to the water. Assuming zero initial vertical velocity, find the average angular velocity during the dive.

A07 [easy]

An object, starting from rest, has an angular acceleration of 1.50 rad/s2. How long does it take to rotate through (a) the first 2.00 rev and (b) the next 2.00 rev?

A08 [average]

A turntable of radius 15 cm starts from rest accelerates with constant angular acceleration to 45 rpm in 2.0 seconds. Find: (a) the angular acceleration, (b) the number of revolutions completed in 5 s, (c) the time needed to complete 2 revolutions, (d) the radial and tangential accelerations of a point on the rim at 1.0 s.

A09 [easy] A plane flies West along the equator just fast enough to keep the setting sun continually in view. How fast is

it flying? [Hint: Relative to the earth, the sun appears to circle us once every 24 hours. Assume that the

plane is not very high above the earth’s surface.]

A10 [average] [extra practice] A rotating object has a constant angular acceleration of 6.0 rad/s2 clockwise. In a certain period of time, it rotates through 6.0 rad clockwise and ends up turning at 9.0 rad/s clockwise.

a) How long did this take? [Hint: You will have to solve a quadratic equation; there will be two solutions.] b) For each of your two solutions, what was the initial angular velocity?

A11 [ easy to average]

A certain car has 60-cm diameter wheels, and does not skid or slip. Their rate of rotation decreases, at a constant rate, from 1200 rpm to 500 rpm in 25.0 s.

a) Find the angular acceleration. b) How much further does the car move before stopping? c) Find the car’s speed when the angular velocity of the wheels is 20 rad/s.

A12 [ challenging]

A wheel A of radius rA = 10 cm is coupled by a belt (B) to wheel C of radius rC = 25 cm. The angular speed of wheel A is increased from rest at a constant rate of 1.6 rad/s2. Find the time needed for wheel C to reach an angular speed of 100 rev/min, assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.)

A13 [ a bit challenging] Find the magnitude of the linear acceleration of an astronaut in these two situations:

a) She is in a centrifuge of radius 15 m that has an angular velocity of 1.2 rad/s and an angular acceleration of 0.8 rad/s2.

b) He is in a space station of radius 1000 m that is rotating at a constant 0.5 rpm.

A14 [average] [extra practice] A turntable of radius 15 cm starts from rest and accelerates uniformly for 2.0 s to reach a final angular speed of 33.33 rpm.

a) Find the angular acceleration. b) How many revolutions does it complete in 5.0 s? c) How long did it take to complete the first 2 revolutions? d) Find the radial and tangential accelerations of a point 6.0 cm from the rim after the first 1.0 sec.

A15 [ easy ] [extra practice]

At t=0 an object is rotating at 50 rpm clockwise. A clockwise external torque gives it a constant acceleration of 0.5 rad/s2 just until it reaches 100 rpm; after that, the external torque is no longer applied. How many revolutions are completed at t=20.0 s?

A16 [ average ] A wooden meter stick is spinning about an axis through the 100-cm mark at the end. The stick lies in the plane of your diagram (draw one!) and the axis is perpendicular to the stick. At t = 0, it was spinning at 55 rotations/sec clockwise. It is slowing down at a constant rate of 10 rotations/s2 . Assume that the angular acceleration remains the same, in magnitude and direction, for the next 8 seconds. Answer each of the following, all in radian measure, for a point A located on the 70-cm mark of the meter stick, at time t = 8 sec. Don’t forget directions, unless the question is only asking for a magnitude.

a) the angular acceleration b) angular velocity c) magnitude of (ordinary, linear) velocity d) magnitude of tangential acceleration e) magnitude of radial acceleration

A17 [ENR] A disk of diameter 12.0 cm is rotating so that the angular position of a radial line given by θ = 10 – 5t + 4t2 radians.

a) Find the average angular speed from t = 1.0 s to 3.0 s. b) Find the linear speed of a point on the rim at t = 2.0 s, c) Find the radial and tangential accelerations of the same point, at the same time.

A18 [average] [extra practice]

A particle moves on a circle of radius r with angular velocity ω and angular acceleration α. Prove that the magnitude of the (total!) linear acceleration is a = r (ω4 + α2)1/2.

A19 [ easy ] [extra practice]

A car with tires of radius 30 cm starts from rest and reaches 108 km/h in 10 s, with no slipping. a) Find the angular acceleration of the wheels. b) Find the number of revolutions completed during this trip. c) At the instant when the car is travelling at 108 km/h, what is the radial acceleration of a point on the rim?

A20 [average] [extra practice]

A wheel has constant angular acceleration, starting from rest. It makes 40 revolutions during the time that the rate of rotation changes from 20 to 50 rpm

a) Find the angular acceleration b) How much time does it take to reach 20 rpm? c) When the wheel reaches 20 rpm, how many revolutions has it made?

B. Torque and Static Equilibrium

Torque

B01. [easy] A thin rod of length L = 8.0 m is pivoted at its center as shown, and acted on by four forces of magnitudes F1 =10.0 N, F2 =15 N, F3 =8.0 N, F4 = 20.0 N.

a) Find the magnitude and direction of the torque by each of the forces. b) Find the net torque on the rod.

B02. [easy]

A bicycle pedal is on a 20-cm rod, inclined at a variable angle θ from the horizontal as shown. You exert a 120 N force vertically downward on the pedal. Find the torque about the axle for the following values of θ: (a) 0; (b) 30°; (c) 45°; (d) 60°.

B03. [ easy ]

The dot is the center of the rectangular object. A force of magnitude F1 = 42 N acts as shown. Find the torque about the pivot P, by the simplest method.

+

L/4

F1 F

2

F3

40o 60o

F4

P

B04. [average] A rod of length L = 1.2 m is pivoted at one end, so that it can rotate in a horizontal plane. Three forces act as shown below; F3 acts at the midpoint of the rod. F1 = 12 N, F2 =5.8 N, F3 = 4.0 N; α = 30o, θ = 200, and β = 37o

a) Find the torque due to each force. b) Find the net torque on the rod

B05. [ qual ] [easy]

You have a choice between two wrenches, one longer and one shorter, to unscrew a stubborn nut. Which will

make your task easier, and why?

B06. [qual ] [ easy-to- average]

As you know, a doorknob is usually at the edge of the door farthest from the hinge. But sometimes a large, massive door has its doorknob in the center of the door. a) Draw a FBD of a door as seen from above. Label the pivot, and show which direction the door will rotate as

you open it. b) Is it easier to open the door if its knob is in the center? Or is it harder? Or just the same as with an ordinary

doorknob? Explain carefully. Your answer does not need to be long, but it should include: state the relevant equation; label the distances in your diagram with useful symbols and subscripts; say which variables are larger, smaller or same in each case; and lead logically to your conclusion.

B07. [easy] [ extra practice]

Three forces, F1 = 5 N, F2 = 2 N, and F3 = 6.6 N, act on a plane body as shown. Find the torque due to each force, about an axis perpendicular to the plane of the diagram passing through point O.

θ

1

F3

F2

L/4

α

o

β

+

F1F

F3

F2

+

140 30o

120o

2m 1.2m

1.5m

B08. [ easy to average]

Three forces of 20 N each are applied to a triangular object as shown. The object is shaped as an equilateral triangle with sides one meter long.

a) Calculate the resultant torque about the point A. b) Calculate the resultant torque about the point B.

Rotational Equilibrium

B09. [easy] A seesaw of length of 4.0 m and mass of 30 kg is pivoted at its center. A child of mass of 25 kg is seated at the left end. a) Where should a 60.0-kg father sit, for the seesaw to be in equilibrium? b) Suppose that the father does not sit on the seesaw, but instead exerts a 350-N force, at a point 50.0 cm

away from the right end. At what angle with the see saw should the force be exerted? c) Find the horizontal and vertical components of the force exerted on the seesaw at the pivot point.

B10. [qual] [easy]

An object suspended by a string will always come to equilibrium with its center of gravity directly below the point of suspension. Why? [ You’ll probably find it helpful to draw a diagram.]

B11. [ partly qual ] [easy]

a) Is the object in the diagram in rotational equilibrium? b) Is it in translational equilibrium? c) Find the net torque on the object, about a pivot at its center. d) Find the net torque on the object, about a pivot at the bottom left corner. What do you notice?

B12. [qual] [average] Two hunters are carrying home a dead deer (yuck) on a long pole. Why is the man at the back smiling?

B

A

10N

10N

20 cm

Your answer should include a well-labelled FBD, equations in symbols, state which variables are larger for each man, and lead logically to your conclusion.

B13. [ average]

A uniform 800-gram rod is 0.50 meters long. It is supported on a pivot 15 cm from the right-hand end. You pull on the rope tied to that end as shown ( ie 10o down from horizontal), and the rod remains at rest as shown (ie at 20o from the horizontal). How hard are you pulling on the rope?

Combined Rotational and Translational Equilibrium B14. [easy]

A uniform rod of unknown length L and weighing 35 N is supported by two columns at its ends as shown. A block of weight 10 N is placed one-quarter of the distance from one end. Find the force exerted by each supporting column the forces exerted by the supports?

B15. [ average]

A hungry bear weighing 700 N walks out on a horizontal beam in an attempt to retrieve a basket of goodies hanging at one end of the beam as shown. The beam is uniform, weighs 200 N and is 6 m long. The basket weighs 80 N. The bear can be treated as a point mass, the beam as a uniform rod.

20°

P

20° 10°

T

Wr 70°

+

a) Draw a free-body-diagram for the beam. b) Find the tension in the wire when the bear is at x = 4 m. c) The left end of the beam is attached to the wall with a hinge. Determine the horizontal and vertical forces

exerted on the beam at the hinge.

B16. [average]

A 60.0-kg diver stands at the end of a 3.0 m board of mass of 10.0 kg. The board is attached to two supports 50.0 cm apart as shown. Neglect the flexing of the board. a) Draw a FBD for the diving board.

b) Find the magnitude and the direction of the force exerted on it by each support.

B17. [ average] A uniform 25-kg pole, 10.0 meters long is attached to a wall by a hinge, and supported by a rope as shown. The rope has tension of 400 N, and makes an angle of 35° with the pole. A small block hangs from a point 2.0 meters from the (left) end of the pole, as shown. The whole thing remains at rest, with the pole tilted at 20° below the horizontal. a) Draw a large, clear free body diagram of the pole. Your diagram must label all relevant information such as

forces, pivot, etc. b) Find the mass of the block. c) (c) Find the vertical and the horizontal components of force applied by hinge on the pole

60

o

X

200

350

Rope

Pole

2 m

Hinge

B18. [challenging]

A car has a mass of 1000 kg, and the distance between front and rear wheels is L = 3 m. The center of mass of the car is at the midpoint between front and rear wheels. A driver and a passenger of combined mass 120 kg are sitting on the front seats at d1 = 1.2 m behind the front wheels, as shown. The back bumper is d2 = 1.3 m behind the rear wheels. The car is parked on a level road.

a) Find the normal force on each front wheel. b) Find the normal force on each rear wheel. c) The driver and passenger get out of the car. Then three students try to lift the back of the empty car by

each applying a vertical force, F, on the rear bumper. What is the minimum force each student needs to apply to lift the back?

B19. [ qual] [ challenging ] You want to pull out a nail from a board. You consider two possible methods: pulling sideways on the handle with force F1, or pulling straight up with force F2, as shown in the diagram. Why is method 1 much easier? Explain carefully.

Your answer should: include an FBD of each case; state the relevant law and equation for each case; label the distances in your diagram with useful symbols and subscripts; say which variables are larger, smaller or same; and lead logically to your conclusion.

B20. [average]

L d2 d1

F1

F2

A uniform rod is 2.0 m long and 10.0 kg. Its lower end is hinged to a wall, and a 200-N load is suspended from the upper end, as shown. A horizontal rope 60.0 cm long joins the center of the rod to the wall. Find the tension in the rope.

B21. [ qual ] [challenging] [extra practice] a) Your “calf muscle” is the big muscle at the back of your lower leg; one end of that muscle is attached behind

your knee, and the other end attaches to your foot at the very back of your heel, as shown. Using estimates from your own body, calculate the tension in your calf muscles when standing on your toes.

b) Your lower jawbone is hinged to your skull at a point very close to your ear. To bite and chew, you contract a vertical muscle which connects the jawbone to your cheekbone (ie near the corner of your eye), as shown. Now, if you want to bite off a piece of hard candy, why is it easier to bite it with your molars (back teeth) than with your front teeth?

B22. [ somewhat challenging]

A 40.0 -kg ladder, 3.0 m long, on a level floor, is leaning against a vertical wall as shown. The ladder is inclined at 53o from the floor. A 50.0-kg person stands on the ladder, two-thirds of the way up. Magnitude of the friction force by the wall is 30.0 N. Everything remains at rest.

a) Draw an FBD of the ladder itself. b) Find all the forces on the ladder by the wall and by the floor (magnitudes and directions).

B23. [ challenging] [ extra practice ]

A “roof truss” is a rigid structure made of several beams (called “members”), used to support a roof. Suppose we have a roof truss of negligible mass constructed as shown below, in which all the horizontal members are 3.0 m long, and the vertical ones are 4.0 m each. It is resting on two columns, one at each end.

53°

The right-hand column can be considered as frictionless. This truss supports 8000 N of “dead load” at its center and is also exposed to 300 N of wind force at the point shown, parallel to that slanting beam.

a) Find the normal and the friction forces on the truss by the left end column, and b) Find the normal force on the truss by the right end column.

B24. [ a bit challenging ] [ extra practice ]

A uniform sphere of mass 0.85 kg and radius 4.2 cm is held in place, touching a frictionless wall, by a massless rope. The rope is attached to the wall at a point lying L = 8.0 cm above the center of the sphere.

a) Find the tension in the rope b) Find the magnitude of force on the sphere from the wall.

300 N

8000 N

300 N

8000 N

3.0 m each

4.0 m

L

r

C. Rotational Dynamics, Moment of Inertia, and Rotational Kinetic Energy

Introduction to rotational dynamics and moment of inertia C01. [ easy ]

If a 32 Nm torque on a wheel causes an angular acceleration of 25 rad / s2 , what is the wheel’s moment of inertia?

C02. [ easy ]

A gymnast launching himself from a springboard has a moment of inertia of 12 kg-m2 about a certain axis. During the launching, his angular speed changes from 0 to 2.0 rotations/sec in 400 ms. (Recall that ms is millisecond, ie 10–3 sec.) a) Find the magnitude of his average angular acceleration. b) Find the magnitude of the average external torque on him, by the board during the launching.

C03. [ qual] [ easy ]

Suppose a person or animal is rotating around an vertical axis passing through the center of her body. Explain briefly ( by a short sentence or two) what happens to her moment of inertia if she: a) curls her body tightly together? b) stretches out her arms and legs?

C04. [ qual] [easy]

True or false, and why: “Two round objects of the same mass and radius must have the same moment of inertia.”

C05. [qual] [ easy]

Consider an object shaped like a wine bottle ( or a bowling pin, or a long narrow cone ). In which of the following situations will its moment of inertia be largest? Which smallest? (i) Axis of rotation is lengthwise, along the bottle’s own axis of symmetry (ii) Axis of rotation is perpendicular to the bottle’s length, and passes through the thin end. (iii) Axis of rotation is perpendicular to the bottle’s length, and passes through the thick end. Explain your answers without using any formulas.

C06. [easy] [ extra practice]

Suppose a constant net torque of 450 Nm cw acts on a rigid object, while the object slows down from 300 rads/s ccw to 20 revolutions/s ccw, in 5.0 s. a) Find the objects moment of inertia. b) If the torque was being caused by a force acting tangentially at a point 25 cm from the axis, how large is

the force? C07. [ qual ] [ average ]

True or False, and explain: “ If an object has a large moment of inertia, that means it is hard to turn. So it will be hard to make it start spinning, and if you stop pushing it will quickly slow down and stop.”

C08. [ average] [ extra practice]

Calculate the rotational inertia of a meter stick, with mass 0.56 kg, about an axis perpendicular to the stick and located at the 20 cm mark. (Treat the stick as a thin rod.)

C09. [ easy to average]

A solid cylinder has diameter 40.0 cm and mass of 10.0 kg. A string is wrapped several times around it, and a constant tension of 4.0 N is applied so that the string unwinds without slipping. There is also a frictional torque of 0.20 Nm acting on the cylinder as it rotates. If it starts from rest: a) how fast is it rotating after 6.0 s? b) How much string has unwound by that time?

C10. [ easy to average] [ extra practice ] A wheel, whose moment of inertia is 0.03 kg·m2, is accelerated by a motor at a constant rate, from rest to 20.0 rad/s in 5.0 s. When the motor is disconnected, the wheel stops in 60 seconds. a) Find the frictional torque. Assume that friction was the same at all times. b) Find the torque applied by the motor during the acceleration.

C11. [easy]

A hollow rubber ball has diameter 1.2 m and mass 3.0 kg. A 400-gram squirrel is clinging to the top of the ball. What is the total rotational inertia of this system, about a horizontal axis through the center of the ball?

Rotational Kinetic Energy and Rotational Dynamics, one body C12. [easy]

Consider an empty paint-can, without its lid, rotating at 2.0 rotations/s about its axis of symmetry. The can has radius of 15 cm. Mass of the bottom is 200 grams, and mass of the “tube” part is 400 grams. Find its rotational kinetic energy.

C13. [ easy ]

A hollow 9.0-kg sphere of diameter 80.0 cm is rotating about a fixed axis through its center. At its “equator”, it is rubbing against a rough surface, causing it to slow down from 7.0 rad/s to 5.0 rad/s. How much heat is produced during this time?

C14. [ average ]

Two solid disks are glued together with a common axis; such a device is often called a “compound pulley”.

The smaller disk has mass of 13 kg and radius 12 cm. The larger disk ( made of a less dense metal) is 7.0 kg and has diameter 40.0 cm. This compound pulley was initially spinning about a vertical axis at 5.0 rad/s, counterclockwise as seen from

above. Then a force �� of constant magnitude P = 2.4 N begins to act at the edge of the large disk; �� continues to act at 70o from the radial line, as shown. a) Find the moment of inertia of this compound pulley.

b) Find the angular acceleration (magnitude and direction) while �� acts.

c) Find the angular velocity after �� has acted for 4.0 sec.

C15. [average]

A long thin bar of length 180 cm is pivoted about a vertical axis through the center (axis is perpendicular to the bar; bar is rotating in a horizontal plane.) A string is attached to one end of the bar, and is pulled so that the string is always at 30o to the bar and has a tension of 120 N. This causes the bar to speed up from 4.0 rad/s to 9 rad/s in 2.0 seconds. a) What is the mass of the bar?

[ Hints for a): Draw the FBD; find torque; find angular acceleration; use rotational 2nd law; use appropriate formula for moment of inertia of this shape ]

b) Consider a point at the end of the bar. Find the magnitudes of tangential acceleration and radial acceleration of this point at t = 0.6 sec.

C16. [average]

Two particles 1 and 2, of equal mass, are attached to the ends of a rigid massless rod. The fulcrum is at a distance L1 = 20 cm from particle 1, and L2 = 80 cm from particle 2. The rod is held horizontally on the fulcrum and then released.

Find the magnitudes of the initial accelerations of a) of particle 1 b) of particle 2

C17. [ average ] [extra practice] A 2.0 kg solid cylinder of radius 40.0 cm and height 25 cm is initially rotating clockwise at 600 rpm on a frictionless axle. A brake begins to push with F = 10.0 N radially inward at the edge as shown in the figure. The coefficient of kinetic friction between brake and disk edge is 0.5. How many rotations does the disk complete before it stops?

F

C18. [ (a) is average, (b) is easy ]

a) A turntable ( solid disk) has a mass of 2 kg and radius of 15 cm. It is initally turning at a constant 33 1/3 rpm, with its motor running. When the motor is turned off, the turntable comes to a stop in 20.0 seconds. What power was the motor supplying to keep the object turning at 33(1/3) rpm?

b) What average power is need to accelerate an object whose rotational inertia is 45 kgm2, from 20.0 rpm to 100.0 rpm in 10.0 seconds?

C19. [ easy to average] [ extra practice ]

Find the rotational kinetic energy of the earth: a) due to it rotation about its own axis b) due to its orbiting around the sun For this question you can approximate the earth as a solid sphere of uniform density. Think about what data you will need; it can be found at the back of most physics textbooks, or on the internet.

C20. [ easy ] [ extra practice ]

(I think the question is fine without a diagram. Or if you want, use the diagram from my solution.) A thin rod of length 11.2 cm and mass 2.4 kg is pivoted at one end, with the axis perpendicular to the rod. Two small massive objects of 0.85 kg each are fastened to the rod, one at the center and one at the end farthest from the pivot. The combined object is turning at 0.3 rad/s a) Find the moment of inertia of this combined object. b) Find its rotational kinetic energy.

C21. [average] [ extra practice ]

Some newer car models save a lot on fuel, by the following method: During braking, the kinetic energy is stored in a massive rotating flywheel instead of dissipating the energy as heat in the brakes. This rotational kinetic energy could then be used to get the car moving again when desired. Consider a 50.0 kg flywheel in the shape of a solid cylinder of radius 0.20 m. a) How fast would it have to rotate (in rads/s) to store all of the kinetic energy of a 1000 kg car travelling at

25 m/s? b) What would be the centripetal acceleration at the rim of the cylinder?

L1 L2

2 1

c) If you doubled the radius, the mass would be 4 times larger. In this case what would be the angular velocity required to store all the necessary translational kinetic energy for the same speed ? What would be the centripetal acceleration at the rim now?

C22. [ easy ] [ extra practice ]

Certain trucks can be run on energy stored in a rotating flywheel, with an electric motor getting the flywheel up to its top angular speed of 200π rad/s. One such flywheel is a solid, uniform cylinder with a mass of 500.0 kg and a radius of 1.0 m. a) What is the kinetic energy of the flywheel after charging? b) If the truck uses an average power of 8.0 kW, for how many minutes can it operate between two

consecutive chargings?

Rotational Kinetic Energy and Rotational Dynamics, problems with several bodies Note to the student: All the questions in this sub-group can be solved using energy ( ie without using dynamics and acceleration.) The ones without springs could also be solved by instead using rotational dynamics and kinematics. C23. [average]

A certain pulley is a solid cylinder of mass M = 2.0 kg and radius R = 0.20 m which can rotate on a frictionless

horizontal axle. A block of mass m = 5.0 kg hangs vertically on a cable wrapped several times around the pulley.

If the block was initially at rest, use energy to find the speed of the block after it has fallen 0.50 meters.

C24. [ challenging] Two blocks with mass M1 = 20.0 kg and mass M2 = 100.0 kg are connected over a pulley (solid disk) of mass 10.0 kg and diameter 80 cm, as shown. M2 is on a frictionless surface. Initially the system is at rest. Use energy principles ( not acceleration calculations) to find: a) The linear speed of the two masses when M2 has travelled 0.50 m b) The angular speed of the pulley at that time. c) Can you find the blocks’ speed after the trip of 0.50 m, if the system was not initially at rest, but had initial

block speed of 1.2 m/s ?

C25. [ challenging]

A frictionless ramp, inclined at θ = 300, has a pulley at the top which is a solid cylinder of mass 20.0 kg and radius of 0.50 m. Two blocks with mass m1 = 30.0 kg and mass m2 = 10.0 kg are connected over the pulley as shown. If the system started from rest, find the linear speed of the blocks and the angular speed of the pulley after m1 moves down by 1.6 meter.(Use energy.)

M2

M1

30°

m1

C26. [ challenging] In the diagram shown, the pulley is a solid disk 8.0-kg disk with diameter 0.40 meters. A block is attached to a spring, of constant k = 24 N/m, by a light rope passing over the pulley. The system starts from rest with the spring unextended. After the block has fallen 50.0 cm, it is travelling at 2.0 m/s. What is the mass of the block?

C27. [ challenging] [ extra practice ]

[Hint: This question might seem to be missing one or two pieces of information. Give them suitable symbols, and carry them through your solution.] A 4.0 kg block is attached to a 32 N/m spring by a rope passing over a solid cylindrical 8.0 kg pulley. Assume no friction. If the system starts moving from rest with the spring unextended, find the speed of the block, and the angular speed of the pulley, after the block has fallen 1.0 m. State each answer as a definite number if possible, otherwise as a function of just the missing variables.

C28. [ challenging] [ extra practice ] A car has four tires, each of mass 25 kg and diameter 60.0 cm and mass 25 kg. Consider each tire as a solid disk. The mass of the car without the tires is 1.0 x 103 kg. a) If the car is travellng at 30.0 m/s, what is its total kinetic energy? b) Starting from an initial speed of 30.0 m/s, the car turns off its engine and rolls up a 10° incline. Neglecting

friction, how far would the car roll along the incline before stopping?

C29. [ challenging] [ extra practice ] A frictionless plane is inclined at 53o. A solid-disk pulley ( 4.0 kg, diameter 1.0 meter) is at the top of the incline. A 2.0 kg block is attached to a light string wound around the pulley.

Use Energy principles ( not acceleration) to determine each of the following, at the instant when the block has slid 2.0 m along the incline: a) The angular speed of the pulley b) The speed of the block. c) Through how many revolutions has the pulley turned?

m60

cm

53o

Rotational Dynamics combined with Translational Dynamics, problems with several bodies C30. [ average]

A certain pulley is a solid cylinder of mass M = 2.0 kg and radius R = 0.20 m which can rotate on a frictionless

horizontal axle. A block of mass m = 5.0 kg hangs vertically on a cable wrapped several times around the pulley.

The block was initially at rest.

Draw ( separate ) FBD’s for the block and the pulley, and find: a) The acceleration of the block, b) The tension in the rope , c) The angular acceleration of the pulley , d) The angular velocity and the angle of rotation after 1.0 sec

C31. [ ENR ] [ average to challenging ] In the diagram shown, a pulley of unknown mass has a radius of 20.0 cm. The horizontal rope has tension 52 N, and a 5.0 kg block hangs on a vertical rope whose tension is 51 N. a) Find the acceleration of the block. b) Find the angular acceleration of the pulley. c) Find the moment of inertia of the pulley.

C32. [ ENR ] [ challenging ]

A frictionless ramp, inclined at θ = 300, has a pulley at the top which is a solid cylinder of mass 20.0 kg and radius of 0.50 m. Two blocks with mass m1 = 20.0 kg and mass m2 = 10.0 kg are connected over the pulley as shown.

Draw clear FBD’s for each mass and for the pulley, and find: a) the tension in each rope b) the ( linear) acceleration of the blocks c) the angular acceleration of the pulley.

30°

m1

C33. [ ENR ] [ average to challenging ] [ extra practice ] Consider the above questions C24, C25 and C29. Be able to solve them without using energy, rather by using dynamics and kinematics.

C34. [ ENR ] [ challenging ] [ extra practice ]

A block of mass 𝒎 = 𝟐. 𝟎𝟎 kg is connected by a massless string to a pulley. The pulley in the figure consists of two disks of different diameters attached to the same shaft. The outer radius 𝑹

is 0.6 m and the radius of the smaller disk 𝒓 is 0.5 m. A constant horizontal force 𝐹 of magnitude 24.0 N is applied to a rope wrapped around the outer disk and the tension in the string is found to be 22.0 N. The pulley is not massless. Draw a clear FBD for the mass and for the pulley and : a) Find the magnitude of acceleration of the block. b) Find the angular acceleration of the pulley. c) Find the moment of inertia of the pulley about its axis of rotation. d) If the block starts from rest, find the kinetic energy of the pulley after 3.0 seconds.

C35. [ ENR ] [ challenging ! ] [ extra practice ]

A solid cylinder of mass M and radius R moves vertically down while a string wrapped around it unwinds on a vertical string. ( somewhat like a yo-yo )

a) Use the energy approach to show that the speed of the spool after it falls a distance h starting from rest

is (4gh / 3)1/2. b) Use the result in part (a) to find the linear acceleration of its center of mass. c) Use dynamics to find the linear acceleration of the spool. d) What is the tension? e) With what force should the string be pulled to have the spool spin but not fall? What is the angular

acceleration in this case?

R

m

R

r

D. Angular momentum D01. [ easy]

Calculate the magnitude of angular momentum for each of the following: a) The exercise ball and squirrel of question C11 ( hollow ball, 3 kg, diameter 1.2 m, squirrel 400 grams on

top, horizontal axis through center) if the ball is rotating at 3.0 rad/s. b) The empty can of C12 ( Radius 15 cm, mass of bottom 200 g, mass of “tube” 400 g, rotating at 2 rots/s). c) The earth, as it spins on its axis. d) The earth, due to its motion around the sun.

D02. [ qual ] [ easy ]

An Olympic diver starts a diving with a jump keeping her body stretched out straight. Just as leaves the diving board, her body is starts rotating slowly counter clockwise. a) To perform a “jack-knife”, during the falling , she holds her body into a fixed roughly triangular shape. What

happens to her rate of rotation, and why? b) Just before hitting the water, she straightens her body again. What effect does this have on her rate of

rotation? D03. [qual ] [ easy to average ] [ extra practice ]

A figure skater starts a spin with arms out, then quickly pulls his arms in close to his body. Use physics principles to explain what will happen now, and why.

D04. [ qual ] [ easy ] Why is a frisbee stable, ie why does the disk keep lying at the same angle while it flies?

D05. [ qual ] [ easy ]

Why is it quite easy to balance on a moving bicycle, but if you sit on it when it’s at rest you’ll probably fall over to one side?

D06. [ qual ] [ easy ] [ extra practice] “Rifling” means there are spiral grooves inside the rifle barrel so that when the bullet comes out, it is spinning about an axis parallel to its length. Explain why this helps to keep the bullet’s nose accurately pointed towards its target.

D07. [ qual ] [ easy to average ]

The term “polar icecaps” refers to the huge amounts of ice near the earth’s north and south poles. Many people are concerned that global warming is melting the polar icecaps. If they do melt a lot, what will happen to the earth’s rate of rotation (day and night) ? Hint: Draw a sketch. If the icecaps melt, where will the big mass of water go?

D08. [average]

A turntable ( solid disk) with radius 15 cm is spinning, on a frictionless vertical axle, at 12 rads/s. A 1.0-kg ball of putty is gently dropped onto the outer rim, making the disk slow down to 8.57 rad/s. Find: a) the mass of the disk b) Was the rotational kinetic energy of the system conserved? If so, how do you know? If not, what happened

to the lost kinetic energy? D09. [average]

A person is spinning on ice at 2 rads/sec about a vertical axis through the center of his body. Approximate the person’s body as a solid cylinder of mass of 80 kg and radius 30 cm. ( In this question we are neglecting the mass of his arms.) Initially he is spinning at 2 rad/s, holding a 6.0 kg block in each hand, at 50 cm from the axis of rotation. Then he stretches out his arms so that each block is to 80 cm from the axis. Consider each block as a point particle. Find: a) The angular velocity of the person when the arms are stretched out. b) The kinetic energy lost/gained, and explain this difference.

D10. [ partly qual ] [average] A person stands near the center of a platform which is rotating in a horizontal plane. She then walks towards the edge of the platform. Treat the person as a point mass and the platform as a massive solid disk. a) With no numbers or calculations ( but with words and equations) explain what will happen to the platform’s

motion, and WHY. b) Suppose the person’s mass is 50.0 kg, the platform is 40.0 kg and has diameter 6.0 m, and was initially

spinning at 10.0 rad/s. The person was initially 0.50 m from the center, and then walks rapidly outwards. Find the angular speed when he has walked 2.0 meters.

D11. [ easy to avg ] [ extra practice ]

A solid disk ( 10.0 kg and radius 50.0 cm) is spinning freely on a vertical axle at 3 rad/sec. A second identical disk slides down the axle, and they lock together and rotate as one object. a) Find the final angular speed of the combined object. b) What is the change in kinetic energy of the system?

D12. [ (a) is average to challenging ] [ b, c are average, extra practice ]

A circular platform (solid disk, 80.0 kg, diameter 4.0 m) is spinning in a horizontal plane at 2.0 rad/s. The axle is frictionless. A 40.0 kg child stands beside the platform. Walking radially, the child steps onto the rim of the disk. a) Does the angular speed of the platform change? If so, why, and what is the new value? b) She then walks to the center. Does the angular velocity change? If so, what is the new value? c) Does the kinetic energy change when she walks from the rim to the center? If so, by how much?

D13. [ slightly challenging ] A thin ring of mass M=1kg and radius R=40.0 cm is spinning about a vertical diameter. ( Notice, NOT about its axis of symmetry. With axis through a diameter, I = 1/2MR2. ) A small bead of mass m = 0.20 kg can slide without friction along the ring. When the bead is at the top of the ring, the angular speed is 5.0 rad/s. How fast is the ring spinning when the bead has slipped halfway to the horizontal (where θ = 45°)?

D14. [ somewhat challenging]

A 100-kg solid cylindrical platform (radius 2.0 m) is initially at rest, with a 60-kg girl standing on the rim. She starts walking on the platform at 2.0 m/s relative to the ground, keeping her direction tangent to the rim in a clockwise sense as seen from above. What is the new angular velocity of the platform ( magnitude and direction)?

D15. [ qual ] [ challenging ]

A person rides without slipping on a platform which is rotating rapidly, clockwise as seen from above. He then starts to walk slowly counterclockwise relative to the platform. While he is walking, does the platform rotate slower, faster, or at the same rate as it did before? Explain, of course, including a diagram.

D16. [challenging] [ extra practice]

A solid disk-shaped platform ( 160 kg, diameter 4.0 m), pivoted on a frictionless vertical axle, is initially at rest. An 80-kg man runs along the tangent towards the platform and jumps onto the rim. Just before landing on the platform, he is moving at 4.0 m/s tangent to the rim. Find the angular velocity of the platform after the man jumps on.

R

M

m

D17. [ qual ] [ easy ] “Neutron stars” are formed after a normal star has contracted due to gravity. Explain why neutron stars tend to have very large angular velocities.

E. Systems of Particles, use of Cross Product, and Rolling To the student: The label ENR means that your own teacher will tell you whether you are responsible for this topic.

Center of Mass of a system of particles [ENR] E01. [ average] [ ENR ] A 2.00 kg particle has the xy coordinates (-1.20 m, 0.50 m), and a 4.00 kg particle has the xy coordinates (0.60 m, -0.75 m). Both lie on a horizontal plane. At what x and y coordinates must you place a 3.0 kg particle such that the center of mass of the three-particle system has the coordinates (-0.5 m, -0.7 m)? x [-1.5m] and y [-1.43m] E02. [ average] [ ENR ]

Consider a system of three particles, with masses m1 = 3.0 kg, m2 = 4.0 kg, and m3 = 8.0 kg. The scales on the axes are set by xs = 2.0 m and ys = 2.0 m. a) Find the x and y coordinates of the system's center of mass.

x[1.1m]and y[1.3m] b) If m3 is gradually increased, does the center of mass of the system shift

toward or away from that particle, or does it remain stationary?

E03. [ easy] [ ENR ] Three particles are located in the xy plane as follows: 2 kg at (- 2, 3) m; 3 kg at (-3, +4) m; and 5 kg at (3, -1) m. What is the position of the center of mass ? [0.2m, 1.3m]

E04. [ average] [ ENR ]

Locate the center of mass of the following molecules, shaped as shown in the diagram: a) In the HCI molecule the atoms are 1.3 x 10-10 m apart. The mass of the H atom is 1 amu and that of the Cl

atom is 35 amu. [0.126 nm from H] b) In the H20 molecule, the mass of the O atom is 16 amu. The H and O atoms are separated by 10- 10 m,

and the bond angle is 105°.[6.76 pm from the O, along the line of symmetry]

y (m)

x (m) 0

m3

m1

m2

xs

ys

O

H

H

105° Cl H

Moment of Inertia of a system of particles ( All ENR except (a,c) of E06.) E05. [ average] [ ENR ]

Four particles in the xy plane are located as follows: 1 kg at (3, 1)m; 2 kg at (-2, 2) m; 3 kg at (l, -1) m; and 4 kg at (-2, -1) m. Find the total moment of inertia :

a) about the x axis [16kgm2] b) about the y axis, [36kgm2] c) about the z axis. [52kgm2]

E06. [ average ] [ parts (b) and (d) are ENR ]

Two particles, 2.0 and 5.0 kg are glued to the ends of a 2.0 meter massless rod. a) Find the system’s moment of inertia about an axis perpendicular to the rod and passing through the mid-

point. [7 kgm2] . b) Find the system’s moment of inertia about an axis perpendicular to the rod and passing through the center

of mass. [5.71 kgm2]. c) .Now suppose that the rod has mass of 1.2 kg. Find the system’s moment of inertia about an axis

perpendicular to the rod and passing through the mid-point. [7.4 kgm2]. d) Again with the rod as 1.2 kg, find the system’s moment of inertia about an axis perpendicular to the rod and

passing through the center of mass. [6.33 kgm2]. E07. [ challenging] [ ENR ]

Two point masses, m1 = 3.0 kg and m2 = 5.0 kg are connected by a zero-mass rod. Find the moment of inertia of the system about each of the axes shown in the diagram. Take d1 = 1.0 m and d2 = 2.0 m.

[IA=23 kgm2, IB=45 kgm2; IC=27 kgm2; ID= 0 ].

E08. [ challenging] [ ENR ] In a water molecule the distance between the oxygen and hydrogen atoms is 9 x 10-11 m and the masses of the atoms are mo = 16mH, where mH = 1.67 x 10-27 kg. The angle between the two H-O bonds is l05o. Find the moment of inertia of the molecule about: a) an axis along the H-O bond [1.26*10-47kgm2] b) an axis through the 0 atom parallel to the line joining the two H atoms [1.0*10-47kgm2]

A B C

D m1 m2

d1 d2

O

H H 105°

Using the Cross Product [ENR] E09. [ average ] [ ENR ]

A particle of mass 4 kg has coordinates x = 2 m, y = -3 m and velocity components Vx = 5 m/s and vy = 7 m/s.

Find its angular momentum about an axis through the origin. [ ]ˆ)/(116 2 kskgm .

E10. [ average ] [ ENR ] For the four particles shown in the diagram: a) Find the angular momentum about the origin of each particle. b) Find the magnitude and direction of the total angular momentum.

]ˆ)/23.0[( kskgm

E11. [ average ] [ ENR ] In the diagram, P is a point in the xy plane, and a 0.400 kg ball is thrown directly upward, at initial 40.0 m/s, from a point 2.0 m to the right of P.

Taking P as the pivot point: a) Find the ball’s angular momentum when it is at maximum height [0]. b) Find the ball’s angular momentum when it is halfway back to the ground [-22.6kgm2/s] c) Find the torque on the ball due to gravity when the ball is at maximum height [-7.84 Nm] d) Find the torque on the ball due to gravity when the ball is halfway back to the ground [-7.84Nm]

E12. [Challenging] [ ENR ]

At a certain instant, a 0.25 kg object, at position mkir )ˆ2ˆ2(

with velocity smki /)ˆ5ˆ5(

. is being

acted on by a force jNF ˆ4

. use unit vector notation to express:

a) the angular momentum of the object. jskgm ˆ)/(5.52[ 2

y (m)

x (m)

2 4 -4

-2

2

4

-4

-2

3 m/s 1.5

m/s

2 m/s

1 m/s

40° 60°

m3 = 4

kg

m2 = 3

kg

m1 = 2

kg

m4 = 5

kg

y

x P

Ball

b) the torque acting on the object . Nmki ]88[

E13. [Challenging] [ ENR ]

A uniform thin rod of length 0.5 m and mass 4.0 kg can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a 3.0 g bullet traveling in the rotation plane is fired into one end of the rod. As viewed from above, the bullet's path makes an angle θ = 60.0° with the rod. If the bullet lodges in the rod and the angular velocity of the rod is 10 rad/s immediately after the collision, what is the bullet's speed just before impact? [1300m/s].

E14. [Challenging] [ ENR ]

In unit-vector notation, express the torque about the origin on a particle located at coordinates (0, -4 m, 3 m), in each of the following cases:

a) if that torque is due to a force with components F1x = 2 N, F1y = F1z = 0 [(6 N · m) j + (8 N · m) k ]

b) if that torque is due to a force with components F2x = 0, F2y = 2.0 N, F2z = 4.0 N [- 22 N · m) i ]

E15. [Challenging] [ ENR ]

A particle located at mkir )ˆ4ˆ3(

is being acted on by a force with components F1x = – 8.0 N F1y =

6.0 N. Find:

a) the torque on the particle about the origin, in unit-vector notation, [50 N · m] k

b) the angle between the directions of r

and 𝐹

Axis

Rolling [ENR] E16. [ average] [ ENR ]

A solid sphere of mass M and radius R rolls without slipping down an incline of angle θ. a) Find the linear acceleration of the center of mass. b) Find the minimum coefficient of friction required for the sphere to roll without slipping.

E17. [ average] [ ENR ]

A marble of radius r rolls without slipping down an incline, and then up along a vertical circular track of radius R. What is the minimum height H from which the ball must start so that it barely stays in contact at the top of the circle? Assume r is very small as compared to both H and R.

A couple of other challenging questions E18. [ ENR ] [challenging] A thin disk of mass M and radius R can rotate freely about a pivot on its rim, as shown. It is released when its center is at the level of the pivot . Derive a formula for the speed of the lowest point of the disk when the center is vertically below the pivot? [(16 g R / 3)1/2]

E19. [ ENR ] [ challenging] The uniform solid block in the diagram has mass 0.172 kg and edge lengths a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Find its moment of inertia about an axis through one corner and perpendicular to the large faces.

.[ 4.7 × 10-4 kg · m2].

pivot

Rotation axis

a

b c