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8.012 Physics I: Classical Mechanics Fall 2008

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics

Physics 8.012 Fall 2007

Final Exam

Monday, December 17, 2007

NAME: _______________S O L U T I O N S__________________

MIT ID number: __________________________________________________

Instructions (PLEASE READ THESE CAREFULLY):

1. Do all EIGHT (8) problems. You have 3 hours.
2. Show all work. Be sure to circle your final answer.
3. Read the questions carefully
4. All work must be done in the white books provided.
5. No books, notes, calculators or computers are permitted
6. A sheet of useful equations is provided on the last page.

Your Scores

Problem Maximum Score Grader 1 10 2 10 3 15 4 10 5 15 6 15 7 10 8 15 Total 100

(d) [2 pts] (Challenging) Consider two uniform disks of mass M, radius R and negligible thickness, connected by a thin, uniform rod of mass M. The centers of the disks are separated by a distance 4R. Reproduce the diagram at right in your solution book and draw the principle axes (^) 4R of this object centered at its center of mass [1 pt], indicating the axis about which torque-free rotations are unstable [1 pt]. Note: you do not need to calculate the moment of inertia tensor to solve this problem.

unstable

axis

⊗

North

(e) [2 pts] A car at latitude λ on a rotating Earth drives straight North with constant velocity v as indicated in the diagram. In (^) West

which direction does friction between the East

car’s tires and the road act on the car to ⊗ counteract the Coriolis force on the car?

R^ R M M

M

λ

East West Coriolis force does not act on the car

Problem 2: Swing Bar Pendulum [10 pts]

M

L/

L

A uniform bar of mass M, length L, and negligible width and thickness is pivoted about a fixed post at a point 1/3 along the length of the bar (see figure). The bar is initially released from rest when it is tipped just slightly off of vertical, causing it to swing downwards under the influence of constant gravitational acceleration g as shown above. Ignore friction.

(a) [5 pts] What is the total force the swinging bar exerts on the fixed post when it passes through horizontal? Express your answer as a vector with components in the coordinate system indicated above.

(b) [5 pts] What is the angular rotation rate of the bar as it swings past it lowest point (i.e., oriented vertically)?

Combining the last two equations yields:

To solve for Fx, we can use conservation of mechanical energy since

only a conservative force (gravity) is doing work on the bar. The kinetic

energy of the bar (translation of center of mass and rotation about center

of mass) is drawn from its gravitational potential energy when it starts at

vertical and its center of mass if L/6 higher:

then using the r equation of motion:

The force that the swinging bar exerts on the fixed post is therefore:

Note that we could have also derived the rotation and energy equations

as pure rotation about the pivot. In this case the rotational equation of

motion, with torque coming from gravity acting at the center of mass,

becomes:

which can be plugged into the θ equation for the center of mass motion

above to obtain Fz. The conservation of energy equation involves only

rotational energy in this case because the pivot is not translating; hence:

as derived above, and this can be plugged into the r equation of motion

to obtain F z as before.

(b) We can use conservation of energy again to compute the angular

rotation rate at the bottom of the swing:

SOLUTION

∆p

α

Mg

N

Ff

(a) There are two solutions for this part based on the ambiguity of

whether ∆p is the impulse imparted onto the ball or delivered by the

stick. Both solutions were accepted. In the former (assumed) case, the

horizontal translational momentum of the ball arises purely from the

horizontal impulse, so that:

The vertical impulse imparts no vertical translation because of the

normal force from the floor; however, it does impart an angular impulse

on the ball:

Alternately, if we consider ∆p to be purely the impulse imparted by the

stick over a short time ∆t, then one needs to also consider the other

forces acting on the ball during that time, in particular friction between

the ball and the floor. The friction force acting on the ball during the

time ∆t is:

where during the time ∆t:

The normal force varies according to the details on how the strike force

acts over the time ∆t. Nevertheless, the change in momentum in the x

direction over this period is:

If we assume that ∆t is very small, then

The angular rotation rate remains the same as above.

(b) For this part assume the first solution for v 0. The ball comes to rest

when it stops rotating and translating. Both motions are retarded by

friction acting at the bottom surface of the ball.

Since these times must be the same we find that:

Problem 4: The Accelerated Atwood Machine [10 pts]

R

M

M

M

An Atwood machine consists of a massive pulley (a uniform circular disk of mass M and radius R) connecting two blocks of masses M and M/2. Assume that the string connecting the two blocks has negligible mass and does not slip as it rolls with the pulley wheel. The Atwood machine is accelerated upward at an acceleration rate A. Constant gravitational acceleration g acts downward.

(a) [8 pts] Compute the net acceleration of the two blocks in an inertial frame of reference in terms of g and A. Do not assume that tension along the entire string is constant.

(b) [2 pts] For what value of A does the block of mass M remain stationary in an inertial frame?

SOLUTION

T 1 T 2

Mg’ T 1 T 2 Mg’/

(a) This problem is straightforward if it is done in an accelerated

reference frame, where the pulley and blocks move under an effective

gravitational field g’ = g + A. The force diagrams of the components of

the system in this reference frame are shown above. The equations of

motion for the pulley and blocks are (with positive z in the direction of

effective gravity):

The massless string implies the constraint equations:

Combining these equations yields:

Problem 5: What is the Best Way to Move a Heavy Load up a Hill? [15 pts]

2 α

μ 1

2M

M

μ 2

2 α

μ 1

2M

M

μ 2

A B

Two students, each of mass M, are attempting to push a block of mass 2M up a symmetric triangular hill with opening angle 2α. Student A pushes the load straight up; student B pulls the load up by running a massless rope through a massless, frictionless pulley at the top of the hill, and pulling on the rope from the other side. The maximum coefficient of friction (assumed here to be equal to the coefficient of kinetic friction) is μ 1 between the students’ shoes and the hill, and μ 2 between the block and the hill. Assume μ 1 > 2μ 2. Constant gravitational acceleration g acts downwards.

(a) [5 pts] For what minimum angle αmin (i.e., maximum steepness) does neither student need to apply any force to hold the load in place?

(b) [5 pts] Calculate and compare the forces each student must exert on the block to move it up the hill at constant velocity. Does either student have an advantage here?

(c) [5 pts] Calculate and compare the minimum angles α < αmin that each student is able to move the block up the hill at constant velocity without their shoes slipping on the hill surface. Does either student have an advantage here?

SOLUTION

2Mg

N μ 1 N

F

2Mg

N

μ 1 N Mg

N’

F μ 2 N’

Mg

F

μ 2 N’

N’

(a) (b) (c) (d)

(a) The force diagram of the boxes when it is not being pulled by the

students is shown in diagram (a) above, for the critical case in which it

almost slips. The diagram is the same for both boxes. The equations of

motion in the z and x directions are:

Note that the way α is defined here, the trigonometric functions are

reversed to what is normally derived.

(b) The force diagram of the boxes when it is pulled with constant

velocity by the students is shown in diagram (b) above, and is the same

for both boxes. For constant velocity, the net force on the block must be

zero, so the equations of motion in the z and x directions are:

Problem 6: Ball Rolling in a Bowl [15 pts]

θ

L

M R

μ

A solid uniform ball (a sphere) of mass M and radius R rolls in a bowl that has a radius of curvature L, where L > R. Assume that the ball rolls without slipping, and that constant gravitational acceleration g points downward.

(a) [5 pts] Derive a single equation of motion in terms of the coordinate θ (the position angle of the ball with respect to vertical) that takes into account both translational and rotational motion, for any point along the ball’s trajectory. Be careful with your constraint equation!

(b) [5 pts] Find the position angle of the ball along the bowl’s surface as a function of time in the case that θ is small. Assume that the ball is started from rest at a position angle θ 0.

(c) [5 pts] At what maximum initial position angle θ 0 can the ball be placed and released at rest and still satisfy the rolling without slipping condition throughout its motion? Note that θ 0 does not have to be small in this case.

SOLUTION

θ

φ

Ff

Mg N



(a) The force diagram for the rolling ball is shown above. Note that

friction points in the direction of rolling, as it acts to reduce the spin of

the ball as it slows down going up the bowl’s incline (if the ball is

rolling down the side of the bowl, friction would also point in this

direction to spin up the ball). Because we are dealing with circular

motion, choose a polar coordinate system centered at the focal point of

the bowl with positive θ increasing as shown. Let φ to be the coordinate

describing the spin of the ball, with positive direction as shown (the spin

vector points out of the page). The equations of motion for translation

of the center of mass and rotation about the center of mass are: