PHY 141 Assignment 5 Summer 2020

Reading: Rotation 2, 3.

Key concepts: Rolling, static equilibrium; torque and angular velocity as vectors; angular momentum and its conservation;

1. A heavy cylindrical container is being rolled up an incline as shown, by applying a force parallel to the incline. The static friction coefficient is . The cylinder has radius R, mass m and moment of inertia about its symmetry axis I.

a. Draw the cylinder and the incline and make a free-body diagram showing all the forces on the cylinder and where they are applied.

b. Assume the motion up the incline is at constant speed, and that . What direction is the static friction force? How do you know?

c. Assuming is large enough, or is small enough, to allow the cylinder

to roll without slipping, what is F? Ans: .

d. Show that the minimum value of F is 1/2 the force one would need to push the cylinder up a frictionless incline of the same angle .

e. Suppose the minimum force is used. What is the largest possible angle of incline, in terms of ? Ans: . [Find the normal force.]

2. The wheel of a stroller needs to be lifted over a curb, as shown. Force F is applied at the axle of the wheel, making angle with the line from the axle to the top of the curb; that line makes angle with the vertical. The wheel has mass M and radius R.

a. Find F in terms of the given quantities.

Ans: .

b. What is the value of for which F is smallest? What is F in that case?

Ans: , .

c. Fo a small wheel, may exceed . What does one do then?

µs

d > R

µs θ

F = Rdmgsinθ

θ

µs tanθ = 2µs

βα

mg sinαsinβ

β

π /2 mgsinα

α π /2

dF

θ

αβ

F

PHY 141 Assignment 5 Summer 2020

3. The weight of the head is supported by an upward force exerted by the atlanto-occipital joint at the top of the neck. But this joint is not directly below the CM of the head, as shown. To hold the head upright, the neck muscles at the back must pull down. (This is why your head falls forward when you doze off.) Take the weight of the head to be 50 N.

a. Find the force N exerted by the joint.

b. Find the force F exerted by the neck muscle when the head is upright.

4. The power supplied by a torque to a system rotating with angular velocity is . You are to show how a car gets its kinetic energy from the torque the engine applies to the drive wheels.

Shown is the car, accelerating to the right. It has total mass M; the wheels (each pair treated as one object) have radius R and moment of inertia I. Its kinetic energy consists of three parts: the CM energy

plus the rotation energy of the front and rear wheels, both . The wheels are not slipping, so . A clockwise torque is applied to the front wheels by the engine. This makes the point of road contact of those wheels tend to slip backwards, so there is a static friction force to the right on them.

a. The rear wheels, not connected to the engine, must also have a clockwise angular acceleration (and therefore total torque) so they don’t slip. What direction is the static friction force from the road on those wheels?

b. The friction forces do two things: they exert torques on their separate wheels and together they give the net external horizontal force that accelerates the CM. Write the three 2nd law equations expressing these things. [Be sure to include in the torque on the front wheels.]

c. Now write the total kinetic energy K and calculate . Use the equations from (c) to show that . This shows that the car’s increase in energy comes from the torque applied to the front wheels, even though it is internal to the car.

τω P = τω

12Mv

2 12 Iω

2

v = Rω τ0

f1

f2

τ0

dK/dtdK/dt = τ0ω

τ0

a

W

N

F

PHY 141 Assignment 5 Summer 2020

5. The bowling ball shown starts down the alley with CM speed , sliding but not rotating. The ball has mass m, radius R, and

moment of inertia about its center . The coefficient of kinetic friction with the alley is . What are the following?

a. The acceleration (magnitude and direction) of the CM while the ball is sliding, and the angular acceleration (magnitude and direction) about the CM during that time. Ans: , to left; , clockwise.

b. The equations for the CM velocity and the angular velocity while the ball is sliding. Ans: ; .

[An animation of this situation is here.]

6. The ball in the previous problem will stop sliding and roll instead when its linear and angular velocities obey the rolling condition. Find the time when that occurs.

a. At what time does that occur? Ans: .

b. What is the CM speed when the ball rolls? Ans: .

c. How far did the ball slide before rolling? Ans: .

7. The refrigerator of mass m shown is being pushed by a horizontal force F applied along a line through its geometric center, which is also its CM. The static friction coefficient between the refrigerator and the floor is . The dimensions of the refrigerator are as shown.

a. If the refrigerator does not tip over, how large must F be to make it slide? Ans: .

b. For what value of F will the refrigerator be just on the verge of tipping over, if it doesn’t slide? [Consider torques about the bottom right corner. The normal force from the floor acts effectively somewhere on the bottom surface of the refrigerator.] Ans: .

c. Draw a line, starting at the CM, along the direction of the vector sum of F and . Show that if that line passes to the right of the bottom right corner, the refrigerator will tip over.

d. For what values of will the refrigerator slide instead? Ans: .

v025mR

2

µk

a = µkg α = 52 µk(g/R)

v(t) ω (t)v(t) = v0 − µkgt ω (t) = 5

2 µk(g/R) ⋅t

t = 2v07µsg

v = 57 vo

x =12v0

2

49µkg

µs

F > µsmg

F = (w/h)mg

mg

µs µs < w/h

•F

w

h

PHY 141 Assignment 5 Summer 2020

8. Suppose the same refrigerator is resting on the bed of a truck that is accelerating to the left. The coefficient of static friction with the bed of the truck is as before. [Use .]

a. Discuss the ways in which this situation is like the one in the previous problem.

b. Find the minimum value of a that will make the refrigerator slide if it doesn’t tip over. Ans: .

c. For what value of a will it be on the verge of tipping over, if it doesn’t slide. Ans: .

9. Two point masses, each of mass m, are attached to a massless horizontal rod of length as shown. They are rotating at angular velocity about a fixed axle perpendicular to the rod, at distance d from its center.

a. What is the moment of inertia of the system about the axle? Ans: .

b. What is the magnitude of the angular momentum? Ans: .

c. If the system consisted of both masses at the CM, rotating about this axle, what would be the angular momentum? Ans: .

d. What is the angular momentum of the actual system about its CM. Ans: . Verify that is obeyed.

10. The same two masses and rod are now made to rotate about the axle which now passes through the CM, but the rod makes angle with the axle.

a. At the instant shown, find the horizontal (x) and vertical (y) components of the angular momentum. [Use for each mass.]

Ans: , where .

b. What is I of the system about the axle? Ans: .

c. Show that .

d. If the axle has no friction it cannot exert a torque along its length. Discuss conservation of total angular momentum for this system. [What is going on with ?]

µs geff

a > µsg

a > (w/h)g

ℓω

m 12 ℓ2 + 2d2( )

m 12 ℓ2 + 2d2( ) ⋅ω

2md2ω

12mℓ

2ω Ltot = rCM ×MvCM +L(about CM)

α

L = r ×mv

Lx = −Lcosα , Ly = Lsinα L = 12mℓ

2 sinα ⋅ω

12mℓ

2 sin2α

L! = Iω

Lx

ω

ℓ

•d

ω

α

•

w

ha

PHY 141 Assignment 5 Summer 2020

11. A child of mass m is placed from rest (gently) at distance r from the axle of the rotating carousel shown from above; The carousel has moment of inertia I about its axle; it was rotating at angular speed before the child landed on it.

a. What was the kinetic energy of the carousel before the child arrived? Ans: .

b. What is conserved in the “collision” of the child and the carousel?

c. After the child comes to rest relative to the carousel, what are the angular

speed and kinetic energy of the system? Ans: ,

d. Now the child crawls toward the axle, stopping at distance r/2 from it.

What is the new kinetic energy? Ans: .

e. Explain the change. [What work was done and by what agent?]

[An animation of this situation is here.

12. The bowling ball of Question 4 can also be analyzed using conservation of angular momentum. Initially it is moving as shown with CM speed , but not rotating, so it is slipping. Friction will slow down its CM speed

and increase its angular speed until it is rolling with . We wish to find these final values.

a. Consider torques and angular momentum about a point on the alley surface, such as P. Show that the total torque due to the forces on the ball is zero about that point.

b. Consider the total angular momentum about that point, using the formula . What is initially, when the ball is not rotating? Ans: , clockwise.

c. What is it finally, in terms of v? [Use the rolling condition and the value of the moment of inertia .] Ans: .

d. Use the result of (a) to determine the final values of and .

ω0

12 Iω0

2

II +mr2

ω0 12

I2

I +mr2ω0

2

12

I2

I +mr2/4ω0

2

v0

vCM ωvCM = Rω

Ltot = rCM ×MvCM +L(about CM) LtotmRv0

I = 25mR

2 75mRv

vCM ω

•

•P

PHY 141 Assignment 5 Summer 2020

13. Another application of the trick used in the previous problem is to a spinning hula hoop. The hoop shown has moment of inertia about its CM. It is thrown to the right with CM speed , and it is initially spinning counter-clockwise with angular speed . After it lands it moves to the right, slipping on the surface. Eventually it rolls with final CM speed v and final angular speed , related by .

a. As in #11, write the total initial angular momentum about a point on the surface. [Take clockwise angular quantities to be positive.]

b. Find the final total angular momentum, and the final CM velocity v.

c. Show that if is large enough, v will be negative, so the hoop will roll backwards.

14. Questions about rotational stability of a vehicle.

a. A truck is shown from the rear as it is rounding a curve. The center of the curve is the the left, indicated by the arrow. The CM of the truck is indicated by the dot. As the speed of the truck is increased, friction with the road is sufficient to keep it from slipping, but it rolls over. Discuss the condition under which this happens, in terms of the direction of in the reference frame of the truck. [Consider torques about the point of contact of the outer wheel with the road.]

b. Let the distance between the wheels horizontally be w and let the CM be at height h above the road. If the radius of the curve is R , what is the maximum speed the truck can have and not roll over?

15. Questions regarding bicycle riding.

a. A bike rider moves to the left as shown The system’s CM is indicated by the dot. He applies his brakes (on the front wheels) to stop quickly, finding himself and his bike suddenly rotated about the point of contact of the front wheel with the road. Explain, using

in the reference frame of the bike.

b. Explain why a bike rider must lean herself and the bike toward the center of a curve to round it. Draw a picture as seen from in front of the bike and consider in the reference frame of the bike.

I = mR2v0

ω0

ω v = Rω

ω0

geff

geff

geff

•

•

ω0v0