A uniform rigid rod rests on a level frictionless surface

CONSERVATION OF MOMENTUM Name INTWO DIMENSIONS

1. Two objects are arranged on a level, frictionless table as shown. Two experiments are conducted in which object A is launched toward the stationary block B. The initial speed of object A is the same in both experiments; the direction is not. The initid and final velocities of object A in each experiment are shown.

The mass of block B is four times that of object A (m" = 4m^).

Top views Velocity Yectors (drawn to scale)

Mech HW-63

{ do,

Experiment 1 - before collision

ld",l= o ilur: ?

- after collision

Tn, . uei

Direction of Lfio

Experiment2 - before collision 2 - after collision

a. In the space provided, draw separate arrows represeiting the direction of the change in momentum vector of object A in the two experiments.

Is the magnitude of the change in m.omcntum of object A in experiment I greater than, less than, or equal to that in experiment 2? Explain.

Experiment I Experiment 2

b. In the space provided, draw separate arrows representing the direction of the change in momentum vector of block B in the two experiments.

Afier the collisions, is the magnitude of the momentum of block B in experiment I greater than, less than, or equal to that in experiment 2? If the momentum of block B is zero in either case, state that explicitly. Explain.

Direction of Al^

Experiment I Experiment 2

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Mech HW-64

Conseroation of momentum in two dimensions

Two objects collide on a level, frictionless table. The mass of object A is 5.0 kg; the mass of object n is f .O tg. The objects stick together after the collision. The initial velocity of object A and the final velocity of both objects are shown.

2.

Before collision dltAi

After collision 1do, = d"r)

(One side of a square represents 0.1 m/s)

In the space provided, draw separate arrows for object A and for object B representing the direction of the change in momentum vector of the object.

Is the magnitude of the change in m,omentum of object A greater than, less than, or equal to that of object B? Explain your reasoning.

Direction of Al Object A Object B

b. System C is the system of both objects A and B combined. How does the momentum of system C before the collision compare to the momentum of system C after the collision? Discuss both magnitude and direction.

Construct and label a vector showing the momentum 6f system C at an instant before the collision. Show your work clearly.

c, Construct and label a vector showing the initial velocity of object B. Show your work clearly.

(Each side of a square represents 0.4 kg'm/s)

(Each side of a square represents 0.1 m/s)

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Consentation of momentum in two dimensions Name

3. Object A collides on a horizontal frictionless surface with an Frictionless horizontal surface

Mech HW-65

initially stationary target, object X. The initial and final velocities of object A are shown. The final velocity of object X is not given.

a, At an instant during the collision, is the net force on object A zero ot nbn-zero?

b. During the collision, is the momentum of object A conserved? Explain.

Is the momentum of the system consisting of objects A and X conserved? Explain.

c. On the same horizontal surface, object C collides with an initially stationary target, objectZ, The initial speeds of objects C and A are the same, ild trtx= trlz) tne.= tltc, After the collisions, object C moves in the direction shown and has the same final speed as object A.

i. In the space below, copy the vectors d6; and d6l with their tails togbther. Use these vectors to draw the change in velocity vector for glider C, AAc.

Top view

Before collision

A X oH

ax at rest

Aftercollision

AX€o 6o,

Arter collision

ii. Is the magnitude of the change in velocity vector of object A greater than, less than, or equal to the magnitude of the change in velocity vector of object C? Explain.

iii. Is the magnitude of the change in momentum vector of object A greater than, less than, or equal to the magnitude of the change in momentum vector of object C? Explain.

iv. Is the final speed of object X greater than, less than, or equal to the final speed of objectZ? Explain.

d. Consider the following incorrect statement: '6liders A ond C have the some chonge in momentum. They hove the some moss, ond because they have the some initiql speed ond same f inol speed, Av is the some for eoch of ,them."

Discuss the error(s) in the reasoning.

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Before collision

Z o

At rest

-

DYNAMICS OF RIGID BODIES Name Mech HW-69

1. Energy analysis of the block-and-spool problem

A block and a spool are each pulled across a level, frictionless surface by a string, as illustrated at right.

The string pulling the block is tied to a small hook at the center of the front face of the block (not shown). The string pulling the spool is wrapped many times around the spool and may unwind as it is pulled.

The block and the spool have the same mass. The strings are pulled with the szlme constant tension and start pulling at the same instant.

Make the approximation that the strings and the hook are massless.

a. Does the spool cross the finish line before, afier, or at the sante instant as the block? Explain.

Tbp view

Start Finish k-- -_-_- d -- *--4

Bloc

Spooll

u Same

b. Consider the following dialogue between two students:

Student l: "f think thot there's the some omount of work done on block ond spool os they ore pulled from the stort to the f inish since they both move the some distonce."

Student 2: 'f disogree. f think thot the hond pullirg the spool does more work thon the hond pulling the block since the string unwinds qs the spool is pulled."

With which student, if either, do you agree? Explin.

When each crosses the finish line, is the total kinetic energy of the spool greater than, Iess than, or equal to that of the block? Explain. (Hint: Use the work-energy theorem.)

d. When each crosses the finish line, is the translational kinetic energy of the spool greater than,less than, or equal to that of the block? Explain.

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Mech Dynamics of rigid.bodies HW-70

2. Three identical rectangular blocks are at rest on a level, frictionless surface. Forces of equal magnitude that act in the same direction are exerted on each of the three blocks. Each force is exerted at a different point on the block (indicated by the symbol "Xo'), as shown in the top-view diagram below. The location of each block's center of mass is indicated by a small circle.

For each of the blocks, draw an affow on the diagram above to indicate the direction of the acceleration of the block's center of mass at the instant shown. If the magnitude of the acceleration of the center of mass of any block is zero, state that explicitly. Explain.

b. Rank the blocts according to magnitude of center-of-mass acceleration, fromlargest to smallest. If any two blocks have the same magnitude center-of-mass acceleration, state so explicitly. Support your ranking by drawinga point free-body diagram for each block.

3, A uniform rigid rod rests on a level, frictionless surface. The diagram below indicates four different combinations of (1) net force on the rod and (2) net torque on the rod about its center of mass. In each box, draw vectors that represent one or two forces that achieve the given combination of net force and net torque. If any combination is not possible, state so explicitly.

For example: In the second case, indicate one or two forces that could be exerted on the rod so that at the instant shown the net force on it is zero, but the net torque on it is not zero.

Tbp view

li,,l = o, l(-,1 = o l{,,*l = o, l(,,1 * o l{,,,1 + o, li,,l = o lF".,l * o, l?".,1 * o

.,,

Block I

Tbp view

Block 2 Block 3

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Dynamics of rigid bodies Name Mech HW-71

4. Three objects of equal mass, A, B, and C, are released from rest at the same instant from the same height on identical ramps. Objects A and B are both blocks, and they slide down their respective ramps without rotating. Object C rolls down the ramp without slipping. Its moment of inertia is unknown.

Objects A, B, and C are made of different materials, thus the coefficients of friction between the objects and their coffesponding ramps are not necessarily the same.

Object A reaches the bottom of its ramp first, followed by objects B and C, which reach the bottom at the same instant.

a. Rank the objects according to magnitude of center-of-mass acceleration,from largest to smallest. If any objects have the same magnitude center-of-mass acceleration, state so explicitly. Explain.

b. Rank the net forces exerted on the three objects according to magnitude, from largest to smallest. If the net force on arly two objects is the same, state so explicitly. Explain.

c. In the spaces provided, draw and label a (point) free- body diagram for each object.

Free-body diagraur for object i\

F-ree-bocly dia*uram

for object B Free-body diagram

for object C

d. Rank the frictional forces exerted on the three objects according to magnitude, from largest to smallest. If the magnitude of the frictional force is the same on any two objects, state so explicitly. Explain your reasoning.

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All three objects are relea.sed

Mech Dynamics of rigiilbodies HW:72

5. Energy analysis of falling-spools experiment

The modified Atwood's machine shown at right consists of two identical spools connected by a massless, ' " inextensible thread that runs over an ideal pulley. The thread is wrapped around spool A many times, but it is attached to a fixed point on spool B, so that spool B will not rotate.

The spools are released from rest from the same height at the same instant.

a. In tutorial, you observed the motion of the spools after they were released. Ignoring small dffirences in their nntions:

. In which direction did each spool move?

. Did spool A hit the ground before, after, or at the same instant as spool B?

Is the mngnitude of the center-of-nutss acceleration of spool A (while it is fallin g) greater than,less than, or equal to that of spool B? Explain.

Is the translational kinetic energy of spool A just before it hits the ground greater thsn, less than, or equal to that of spool B? Explain.

d. Is the total kinetic energy of spool A just before it hitsothe ground greater than, less than, or equal to that of spool B? Explain.

Consider the system consisting of all of these objects: spool A, spool B, the thread, the pulley, and the Eafth.

i. Explain how you can tell that the total energy of this system(i.e., Ugnv,o+ Ug^",n * Kton,,e * K,*nr. n * Kro' a, * K,o' s) is constant as spools A and B fall.

b.

c.

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Dynamics of rtgtd bodies Narne Mech HW-73

Suppose that this system starts with Ugo", o = f/srr, B = 9 J. Just before the spools hit the ground, which is where the zero for gravitational potential energy is chosen, spool A has translational kinetic energy K,on,"a = 4 J. Determine the value of the rotational kinetic energy of spool A at this instant. Show your work.

6. A third identical spool, spool C, is added to the falling-spools experiment described in the preceding problem.

As above, all spools are released from rest from the same height at the same instant. Spool C is not in contact with any other objects as it falls.

a, Rank the spools according to magnitude of center -of-mas s acc eleration (while falling), from largest to smallest. If any spools have the same center-of-mass acceleration, state so explicitly. Explain.

b. As in the preceding problem, suppose that Ugrrr. n = Ugr,, s = 9 J before the spools are released. Just before the spools hit the ground, which is where the zero for gravitational potential energy is chosen, spool A has trarrslational kinetic energy K,*,, e,= 4J.

i. Rank the spools according to maximum tanslatianal kinetic energy, from largest to smallest. If any spools have the same maximrrtn translational kinetic energy, state so explicitly. Explain. (Use the definition K*, = lma"^'.)

ii. Rank the spools according to maximum total kinetic energy, from largest to smallest. If any spools have the same maximum total kinetic energy, state so explicitly. Explain.

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CONSERVATION OF ANGULARMOMENTUM

Name Mech HW-77

1. In tutorial, you observed the following three experiments involving a student sitting at rest on a stool, holding a spinning bicycle wheel as shown at right:

Experiment 1: The student places his arm against the side of the wheel, slowing it to half its initial angular speed.

Experiment 2: The student places his arm against the side of the wheel, bringing it to a stop.

Experiment 3: The student quickly flips the wheel over (so that it is spinning clockwise when viewed from above, with the same angular speed it had initially).

Student initially at rest

Initial sense of wheel's

rotation

a. You observed that the final angular speed of the student in experiment 3 is greater than that in experiment2, Account for this result using the ideas developed in the tutorial.

b. Rank the experiments according to final kinetic energy of the wheel, from largest to smallest. If the final kinetic energy of the wheel is the same in any two experiments, state so explicitly. (Hint: Can kinetic energy ever be negative?) Explain.

Rank the experiments according to final kinetic energy of the student, from largest to smallest. If the final kinetic energy of the student is the same in any two experiments, state soexplicitly. Explain.

d. Rank the followingfour quantities from largest to smallest: the initial kinetic energy of the wheel (K*i) and the final kinetic energy of the student-wheel system in experiments 1 ,2, and 3 (Kstr, etc.). Explain. (Hint: It may be helpful to think about changes in energy other than mechanical energy.)

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HW-78 Conseruation of angular momentum

Z. The diagram below illustrates four hypothetical collisions that take placg on a level, frictionless surface.- The collisions are shown from a top-view perspective. All pucks are identical. If a linear or angular velocity is not specified, it is zero. If distances appear to be equal, assume that they are.

For each hypothetical collision:

a. Specify the direction of the angular momentum of the rod-puck(s) system with respect to the center of the rod both before and after the collision. If necessary, use the convention that a vector into the page is represented by the symbol I and a vector out of the page is represented by the symbol O.

b, Specify the direction of the linear momentum of the rod-puck(s) system, both before and after the collision.

c. On the basis of your answers above, state whether each hypothetical collision could ot could not occrfi. If a particular hypothetical collision could not occur, state whether it violates (l) the principle of conservation of linear momentum, (2) the principle of conservation of angular momentum, or (3) both.

Before collision

Case I

Case 2

Case 3

Case 4

Ball sticks to

rod

Sense of rotation

After collision

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