AP Physics Assignment – Rotational Mechanics

Objectives/HW

 The student will be able to: HW: 1 State and apply the relations between angular position, angular displacement, angular speed, angular velocity, and angular acceleration to solve related problems. 1 – 3 2 State and apply the relations between the angular (or rotational) motion of a body or system and the linear (or translational) motion of a point on the body or system. 4 – 7 3 Determine the torque of an applied force and solve related problems. 8 – 12 4 Determine the moment of inertia for a system of masses or sold body and solve related problems. 13 – 18 5 State and apply Newton’s 2nd Law for fixed-axis rotation to solve related problems. 19 – 21 6 Apply work and energy to solve fixed-axis rotation problems. 22 – 25 7 State and apply Newton’s 2nd Law for rolling (rotation and translation) to solve related problems (including those with slipping and without slipping) 26 – 33 8 Apply work and energy to solve rolling problems. 34 – 36 9 Determine angular momentum for a particle, system, or rotating body and relate to torque and angular impulse to solve problems. 37 – 42 10 Apply conservation of angular momentum to solve related problems. 43 – 49

Homework Problems

1.      An old record turntable is set for 45.0 rpm.  Starting from rest it takes 1.50 seconds for the turntable to attain this rate.  (a) Determine the angular speed of the turntable when it turns at 45.0 rpm.  (b) Find the rate of angular acceleration when it is first turned on.  (c) Find the angular displacement of the turntable as it accelerates.  How many “turns” is this?

2.      The same turntable is now switched to play an LP it slows at a rate of 2.00 rad/s2 from 45.0 rpm to 33.3 rpm.  (a) Find the time for this to occur.  (b) What is the angular displacement during this interval?

3.       A good-ol’ boy at a stoplight revs the engine of his 426-Hemi ‘Cuda such that the tachometer goes from 1000 rpm to 4000 rpm in 0.50 s.  (a) Find the average angular acceleration of the engine’s crankshaft.  (b) Determine how many revolutions occur.

4.      Starting from rest, a Ferris wheel of diameter 30.0 m undergoes an angular acceleration of 0.0400 rad/s2.  A certain rider is at the lowest point of the wheel just as it starts to move.  (a) Find the velocity of the rider just as he completes a quarter of a turn.  (b) Find the radial and tangential components of his acceleration at the same point.  (c) How much farther must the wheel turn before the rider attains a speed of 6.00 m/s (the maximum that occurs during the ride)?

5.      A helicopter rotor has a total length of 7.50 m and turns at a rate of 47.0 rad/s CW and has an angular acceleration 1.5 rad/s2 CCW.  For a point at the tip of the rotor, find the (a) speed, (b) radial acceleration, (c) and tangential acceleration.

6.      A cyclist is traveling at 7.0 m/s to the right when she applies the brakes and slows to a speed of 5.0 m/s.  Each wheel of the bicycle has radius 0.33 m and completes 5.0 revolutions during this braking period.  (a) Find the time that elapses.  (b) Determine the angular acceleration of each wheel.  (c) Determine the linear acceleration of the bicycle.

7.      A string is wrapped around the axle of a gyroscope – 30.0 cm of the string is in contact with the axle, which has diameter 2.20 mm.  Starting at rest the string is pulled with a constant acceleration, which causes the gyroscope to start spinning.  It takes 1.10 seconds to pull the string off of the axle and the gyroscope then spins for an additional 60.0 seconds before stopping.  (a) Find the maximum angular speed of the gyroscope in rad/s and rpm.  (b) What are the rates of angular acceleration and angular deceleration?  (c) Find the total number of revolutions the gyroscope will spin.

8.      As shown below a square metal plate with sides of length 20.0 cm is subject to three forces:  F1 = 60.0 N, 180.0°, F2 = 90.0 N, 30.0°, F3 = 100.0 N, 180.0°.  (a) Find the torque of each force about the lower left corner of the square.  (b) Find the torque of each force about the midpoint of the upper side of the square.  (c) What additional horizontal force applied at the lower left corner would make the net torque equal zero about this midpoint?

9.      The pedal of a bicycle has a radius of 0.17 m relative to the center of the crank.  A rider of mass 80.0 kg puts all of his weight on the pedal as it moves from top to bottom.  Find the torque on the crank when the pedal is in each of the following positions:  (a) r = 0.17 m, 90.0°, (b) r = 0.17 m, 45.0°, (c) r = 0.17 m, 0.0°.

10.  The nut on the drive axle of a 1974 VW bus must be torqued to 407 Nm.  In order to apply this much torque the mechanic attaches a wrench to the nut and places a “cheater” pipe over the wrench in order to “lengthen” it.  (a) If length from the center of the nut to the end of the cheater is 0.500 m, what is the minimum force that must be applied?  (b) If the mechanic is only capable of exerting 445 N of force how long must the cheater be?  (c) Draw a picture showing the most effective way to arrange the cheater and apply the force.

11.  A 1250 kg car is stopped at a red light on a hill of incline 10.0°.  (a) Assuming the weight is equally distributed on the four tires of radius 30.5 cm each, what is the torque due to friction about the center of each wheel?  (b) This torque must be “countered” by the brakes to prevent the car rolling.  If the car has disk brakes and the calipers are located 11.4 cm from the center of the wheel, what is the amount of tangential force on the calipers?

12.  A meter stick of mass 80.0 grams is balanced vertically on one end, but then it falls over.  Find the torque about the lower end of the stick (a) when it is tilted 45.0°, and (b) right before it hits the ground.

13.  In a physics demonstration a student stands upright, centered on a platform that can rotate freely.  The student holds in each hand a 0.500 kg mass.  Find the moment of inertia of the two masses about the axis of rotation when (a) the two masses are held at the shoulders and are 0.50 m apart, and (b) the two masses are held at arms length and are 1.50 m apart.

14.  Compact masses of 4.00 kg and 5.00 kg are attached to the ends of a rod of length 90.0 cm and negligible mass.  (a) Find the moment of inertia about the center of the rod.  (b) Find the moment of inertia about each end of the rod.  (c) Find the moment of inertia about the center of mass.  Assume the axis is perpendicular to the rod in each case.

15.  Use reference Table 8.1.  (a) Find the moment of inertia of a solid disk of mass 3.00 kg and radius 0.500 m about an axis passing perpendicularly through its center.  (b) Repeat for a hoop of the same mass and radius.  (c) Repeat for a solid sphere of the same mass and radius.

16.  Repeat part (a) of problem 14 using the diagram below and taking into account the masses are solid spheres of radius 10.0 cm and 11.0 cm and the rod has mass 250 grams.  Hint: use the parallel axis theorem for the spheres!

17.  Find the moment of inertia of the thin triangular plate of mass M about the y-axis.

18.  Use the parallel-axis theorem to find the moment of inertia of the triangular plate from the previous problem about a vertical axis that passes through its center of mass.

19.  A uniform cylindrical grinding wheel of mass 5.00 kg and radius 20.0 cm is turned by an electric motor.  Once it is turned on it takes 10.0 seconds for the wheel to go from rest to 125 rpm.  Ignore friction in the bearings of the wheel.  (a) Find the amount of torque exerted on the wheel by the motor.  (b) The motor is switched off and a piece of metal is pressed against the edge of the wheel with a force of 45.0 N.  If the coefficient of friction is 0.500 how many turns will the wheel complete before coming to a stop?

20.  An Atwood’s machine is formed by a string passing over a solid pulley of mass 50.0 grams and diameter 5.00 cm.  The masses on each end of the string are:  m1 = 200.0 g, and m2 = 300.0 g.  Friction is negligible.  (a) Find the angular acceleration of the pulley as one mass falls and the other rises.  (b) Find the linear acceleration of m1.  (c) What would be the linear acceleration of m1 if the pulley were “massless”?

21.  A teeter-totter (or “seesaw”) of length 3.00 m and mass 50.0 kg is pivoted at its center.  A boy of mass 40.0 kg sits 1.40 m left of center and a girl of mass 30.0 kg sits 1.30 m right of center.  Ignore friction in the bearing.  For the instant when the teeter-totter is horizontal find:  (a) the angular acceleration, and (b) the linear acceleration of each child.  (c) As the teeter-totter tilts past horizontal, do the acceleration rates increase, decrease, or stay the same?  Explain.

22.  A string passes over a pulley of mass m and radius r and connects a cart, mass M1 and a hanging mass M2 as shown in the diagram.  The pulley is a solid disk and friction is negligible.  (a) Solve for the linear acceleration of the cart once the hanging mass is released.  (b) If an experiment is done with this arrangement and it is assumed that the pulley is massless, what will be the effect on the results?  What type of error?  Explain.

23.  A stick of length L is held vertically with one end at rest on the floor.  The stick is released and it rotates around its bottom end until the top end hits the floor.  The bottom end does not slip on the floor.  Find the speed and acceleration of the top end of the stick just before it hits.

24.  A cylinder of radius 0.200 m with a moment of inertia of 5.00 kg m2 is mounted on a horizontal axle with negligible friction.  A string is attached and wrapped several times around the rim of the cylinder, which is initially at rest.   A constant force of 30.0 N is applied to the end of the string and it is pulled 4.00 m.  (a) Find the angular speed attained by the cylinder.  (b) Suppose the experiment is repeated but the applied force is replaced with a hanging weight of 30.0 N and it falls 4.00 m starting from rest.  Again find the angular speed attained.

25.  In order to do a certain demonstration, a physics instructor uses his hand to spin a bicycle wheel of diameter 0.500 m.  The wheel has a mass of 1.50 kg, most of which is in the rim.  He grabs the rim and pulls it through precisely one half of a turn to get it spinning at 4.00 Hz.  The wheel continues to spin for 450 revolutions before it stops.  (a) Find the maximum kinetic energy of the wheel.  (b) Find the net work required to set it into motion.  (c) Find the force the hand exerts on the wheel.  (d)  What is the torque due to friction in its bearings?

26.  A solid cylinder rolls down a ramp inclined at angle θ above horizontal.  Determine its acceleration.

27.

 F

A spool has properties:   inner radius = r, outer radius = R, mass = M, and moment of inertia I = MR2/3.  A string is wrapped around the inner cylinder and a force F is applied to the end, as shown in the diagram below.  The spool rolls without slipping.  Find the acceleration, including direction.

28.  Repeat the previous problem, but now the string is pulled straight upward with force F.

29.  A string is wrapped around the circumference of a solid cylindrical disk of mass M and radius R.  The string is held by a person and the disk is released so that it falls as the string unwinds (like a yo-yo).  (a) Find the downward acceleration of the disk if the string does not slip.  (b) Repeat for the same situation but this time the object is an “empty” cylindrical shell (like a can with the ends cut out).

30.  A solid sphere of mass 4.00 kg and radius 10.0 cm is pushed across a horizontal surface.  The force is applied by pushing horizontally on it with a wood block.  Friction for all surfaces can be modeled by: μs = 0.40 and μk = 0.30.  (a) Find the linear acceleration of the sphere if the normal force of the wood on the sphere is 10.0 N, 0.0°.  (b) What is the maximum normal force that the wood can exert on the sphere without slippage on the horizontal surface?  (c) If this is just barely exceeded, what is the angular acceleration of the sphere?

31.  A yo-yo of diameter 6.00 cm is spinning freely on the end of its string and is hovering just above the floor when the string breaks.  At that instant it is on the floor spinning at 175 rad/s but not yet moving across the floor.  As it spins, friction causes it to accelerate across the floor as it continues to slip; μk = 0.250.  Eventually it rolls without slipping on the floor.  Use an appropriate model for moment of inertia.  (a) Find the linear acceleration.  (b) Find the angular acceleration.  (c) Determine the eventual speed attained by the yo-yo.

32.  A bowling ball is thrown straight down the alley with speed vo.  Initially the ball is not spinning, but as it slides across the floor it starts to spin.  For a brief period it continues to slide as it spins faster and faster.  Eventually it is rolling without slipping.  (a) Find the angular acceleration of the ball as it slips (in terms of g and μk).  (b) Find the linear acceleration as it slips.  (c) Find the distance it moves before it rolls without slipping.  (d) Find the final speed of the ball in terms of its initial speed.

33.  A unicyclist applies a downward force to the front pedal when it is in a horizontal position (and no force on the rear pedal).  Find the forward acceleration in terms of:  M = mass of person and unicycle frame, m = mass of wheel, r = radius of wheel, F = downward force applied to pedal, l = length of pedal crank.  Hint:  analyze external force(s) on the system of masses and also analyze torque(s) on the wheel.  Ignore friction in the bearing of the wheel.

34.  A marble of mass 20.0 g rolls across the floor at 5.00 m/s and then up an incline of 30.0°.  (a) Find the kinetic energy of the marble.  (b) How far up the ramp will it roll?  (c) Find the speed of the marble when it is halfway back down the ramp.

35.  A solid steel ball of radius r is placed on a track that leads to a loop-the-loop of radius R.  See the diagram below.  (a) Determine the minimum value of h from which the ball can be released and it will roll without slipping all the way around the loop.  (b) Find the speed of the ball as it exits the loop on a horizontal path.

36.  Inspect the diagram below.  The solid cylindrical disk of mass 2.0 kg is pulled across a table by a falling mass of 0.50 kg.  The string is wrapped around the disk.  Ignore friction and mass of the pulley.  (a) Find the acceleration of each object.  (b) Find the speed of each object after the falling mass has moved downward 0.30 m (starting from rest).

37.  An airplane of mass 3500 kg flies with velocity 75 m/s, north at altitude 750 m above ground.  (a) Find the angular momentum of the airplane about a point on the ground directly beneath it.  (b) Would this value change as the plane moves away from the same point?  Explain.

38.  A ball of mass 175 g is attached to a string and it is twirled around in a horizontal circle of radius 75.0 cm at a frequency of 2.00 Hz.  It revolves clockwise as seen from above.  (a) Find the magnitude of its linear momentum.  (b) Find its angular momentum vector about the center of the circle.  (c) If the string breaks and the ball flies off through the air, explain the effect on its angular momentum, if any.

39.  A mass of 3.00 kg is moving with velocity 2.00 m/s, 30.0° when it is at a position of (0 m, 4.00 m) in the xy plane.  (a) Find its angular momentum about the origin.  (b) Find its angular momentum about the point (−5.00 m, 4.00 m).

40.  A door has mass 10.0 kg and dimensions 0.900 m × 2.15 m.  It is hinged with negligible friction along one of the long edges.  Suppose a kid shoves the door exactly in its center with an average force of 10.0 N normal to the surface for 0.50 s.  (a) Find the magnitude of the angular impulse.  (b) Find the resulting angular speed of the door.  (c) Now suppose the door’s swinging motion is stopped in 0.10 s by a bumper that is hit by the handle, which is located 0.800 m from the hinge.  What is the average normal force on the bumper?

41.  An 80.0 g meter stick is at rest on the floor when a kid kicks it at the 90.0 cm mark.  He delivers a rapid impulse of 0.300 Ns, north in a direction perpendicular to the stick.  Ignore friction with the floor.  (a) What is the resulting linear momentum of the stick?  (b) Find the speed of the center of mass of the stick.  (c) Find the angular impulse about the center of mass.  (d) What is the resulting angular momentum of the stick’s rotation?  (e) Find the angular speed of the stick.  (f) What are the instantaneous velocities of each end the stick?  Hint: use velocity relative to CM to find velocity relative to floor.

42.  Suppose the kid in the previous problem kicks the stick at the center of percussion or “sweet spot”, which is at a point two thirds of the length from one end.  Show that for any mass, length, or amount of force, one end of the stick will have an instantaneous velocity of zero immediately after the impact.

43.  A solid disk of mass 200.0 g and radius 10.0 cm is rotating freely about a vertical axis through its center at 20.0 rpm.  A ring of mass 100.0 g and radius 5.00 cm that is not rotating is centered above the spinning disk and dropped onto it.  The ring “sticks” to the disk such that both objects rotate at the same rate.  (a) Find the resulting angular speed.  (b) What amount of angular momentum is transferred?  (c) What is the change in kinetic energy of the system?  What energy transformation occurs to explain this change?

44.  An ice skater starts to spin about a point on the ice and begins rotating 2.00 times per second.  Initially the moment of inertia of the skater is 5.00 kg m2, but this is reduced to 2.1 kg m2 as the skater draws his arms and legs inward toward the axis of rotation.  (a) Find the angular momentum of the skater.  (b) Find the final rotation rate of the skater.  (c) Find the change in the kinetic energy of the skater.  (d) Based on the work-energy principle work must be done to account for a change in kinetic energy – explain how and what work is done in this case.

45.  The speed of Earth is 30.3 km/s when it reaches its perihelion distance of 1.47 × 1011 m.  What is the speed of Earth when it reaches its aphelion distance of 1.52 × 1011 m?

46.  Pluto orbits the Sun in an elliptical orbit.  The distance from the Sun varies from 4.437 × 1012 m to 7.376 × 1012 m.  Use conservation of energy and conservation of angular momentum to solve for the speed of Pluto at these two points.  Mass of Sun = 1.989 × 1030 kg.

47.  A playground merry-go-round, initially at rest, has a diameter of 4.00 m and a mass of 500.0 kg.  A 40.0 kg kid comes running at it along a tangent line at speed 8.00 m/s and jumps on to its edge and hangs on.  (a) Approximate the rotational inertia of the merry-go-round as a solid disk in order to find its resulting angular speed.  (b) What mass kid performing the same maneuver (running at the same speed) would cause it to rotate at 0.75 rad/s?

48.  A thin rod of mass 300.0 g and length 90.0 cm is mounted on a horizontal axle (with negligible friction) that passes through its center.  The rod is initially at rest in a vertical position.  A 50.0 g lump of clay is launched at the rod with velocity 10.0 m/s, 0.0°, hitting and sticking to the rod at a point 35.0 cm above the axle.  (a) Find the angular momentum of the clay about the axle.  (b) Find the angular velocity of the rod and clay immediately after impact.  (c) Find the linear velocity of the clay after the rod has turned 180.0°.  (d) If the experiment is repeated, but this time the clay hits below the axle, what is the minimum distance below the axle at which the impact of the clay would cause the rod to spin 180.0° in the opposite direction?

49.  An ice cube with sides of length L is sliding at speed v with very little friction across a level countertop.  At the edge of the counter is a small ridge that stops the sliding of the cube.  If the cube is moving fast enough it will rotate “up and over” the ridge and fall off the table.  Given the moment of inertia of a solid cube around one edge is 2ML2/3, show that the minimum speed that will make it fall off the table is approximately:  .  Hint:  use conservation laws.

2.      a. 0.613 s

b. 21

4.      a. 5.32 m/s, up
b. 0.600 m/s2, up
1.88 m/s2 toward center
c. 24.6°

5.      a. 176 m/s
b. 8280 m/s2 toward center
c. 5.63 m/s2, opp. of the rotation

6.      a. 1.7 s
c. 1.2 m/s2, left

7.      a. 496 rad/s; 4740 rpm
c. 2410 rev

8.      a. τ1 = 12.0 N m, CCW
τ2 = 6.59 N m, CW
τ3 = 0.0 N m
b. τ1 = 0.0 N m
τ2 = 4.50 N m, CCW
τ3 = 20.0 N m, CW
c. 77.5 N, 0.0°

9.      a. 0.0 N m
b. 94 N m
c. 130 N m

10.  a. 814 N
b. 0.915 m
c.

11.  a. 162 N m
b. 1420 N

12.  a. 0.277 N m
b. 0.392 N m

13.  a. 0.063 kg m2
b. 0.56 kg m2

14.  a. 1.82 kg m2
b. 4.05 kg m2; 3.24 kg m2
c. 1.80 kg m2

15.  a. 0.375 kg m2
b. 0.750 kg m2
c. 0.300 kg m2

16.  1.88 kg m2

17.

18.

19.  a. 0.131 N m
b. 0.303 turn

b. 1.87 m/s2, up
c. 1.96 m/s2, up

b. 1.40 m/s2, down (boy)
1.30 m/s2, up (girl)
c. decrease – why?

22.  a.
b.

23.

25.  a. 29.6 J
b. 29.6 J
c. 37.7 N, tangentially
d. 0.0105 N m

26.

27.

28.

29.  a. 2g/3
b. g/2

30.  a. 1.3 m/s2, right
b. 41 N

31.  a. 2.45 m/s2, gaining speed
b. 163 rad/s2, losing angular speed
c. 1.75 m/s

32.  a.
b.
c.
d.

33.

34.  a. 0.350 J
b. 1.79 m up (3.57 m along)
c. 3.54 m/s

35.  a.
b.

36.  a. 2.0 m/s2 (disk), 3.9 m/s2 (other)
b. 0.77 m/s (disk), 1.5 m/s (other)

37.  a. 2.0 × 108 kg m2/s, west
b.

38.  a. 1.65 kg m/s
b. 1.24 kg m2/s, down
c.

39.  a. 20.8 kg m2/s, CW
b. 15.0 kg m2/s, CCW

40.  a. 2.3 N m s
c. 28 N

41.  a. 0.300 kg m/s, N
b. 3.75 m/s
c. 0.120 N m s
d. 0.120 kg m2/s
f. 12.8 m/s, N (100 cm mark)
5.25 m/s, S (0 cm mark)

42.

43.  a. 1.68 rad/s (16 rpm)
b. 4.19 × 10−4 kg m2/s
c. −0.439 mJ – explain!

44.  a. 62.8 kg m2/s
c. +550 J
d.

45.  29.3 km/s

46.  6112 m/s, 3677 m/s

b. 58 kg

48.  a. 0.175 kg m2/s, CW