AP Physics C: Mechanics – Systems of Particles and Momentum

 

Reading            Physics for Scientists and Engineers  – pp. 296 - 298, Chapter 11

 

 

Objectives/HW

 

 

The student will be able to:

HW:

1

Determine the center of mass for a set of objects or particles and/or a continuous distribution of mass.

1 – 7

2

Apply Newton’s 2nd Law to a system of particles and solve related problems either with the presence or absence of external forces.

8 – 12

3

State and apply the Law of Conservation of Momentum and solve related problems.

13 – 23

4

Define and apply elasticity and solve related problems.

24 – 30

5

Define and apply the concept of impulse and solve problems that relate momentum, force, and impulse.

31 – 38

6

Solve problems involving variable mass such as that of a rocket.

39 – 40

 

 

 

Homework Problems

 

1.      Two objects are located on the x-axis:  mA = 2.0 kg, xA = 3.0 m; mB = 7.0 kg, xB = 4.0 m.  (a) Find the center of mass of the two objects.  (b) An additional mass mC = 5.0 kg would need to be located where in order to move the center of mass to the origin?

2.      Determine the location of the center of mass of the Earth-Moon system.  The masses are 5.974 × 1024 kg, 7.35 × 1022 kg and the distance between centers is 384000 km.

3.      Two kids sit at the ends of a teeter-totter; one is 50.0 kg and the other is 35.0 kg.  The teeter-totter is 3.00 m long and has a mass of 10.0 kg.  Determine the location of the center of mass of the three objects (this is where the pivot would need to be located so that it would be balanced).

4.      A certain javelin has a length of 2.20 m and is tapered such that its mass per unit length varies according to the following equation:  λ = ax3 + bx2, where a = −512 g/m4, b = 1130 g/m3, and x is measured from the tail of the javelin toward the head.  (a) Find the mass of the javelin in grams.  (b) Find the center of mass of the javelin.

5.      Show that the center of mass of the uniform triangular plate shown in the figure below is located at x = b/3, y = h/3.

6.      A trapezoid has its longest base, b1, along the x-axis and one corner at the origin.  The rest of the trapezoid is in the first quadrant.  Find a formula for the y-coordinate of the center of mass.  Hint: use the previous result for the triangle as an aid.  Also helpful:  the area of a trapezoid is A = ½ h(b1 + b2).

7.      A circular hole of radius R is cut out of a rectangular plate as shown in the diagram below.  The center of the hole is at (c, h/2).  Determine the location of the center of mass of the plate.

8.      Two carts, m1 = 3.00 kg and m2 = 2.00 kg, are connected by an elastic cord and are at rest on a frictionless surface.  The two carts are initially 1.00 m apart and all the slack is out of the cord (but the cord is not stretched).  A constant force horizontal force of 10.0 N is applied to the smaller cart, pulling it away from the larger cart.  (a) Find the initial position of the center of mass.  (b) Find the acceleration of the center of mass.  (c) How far has the center of mass moved and what is its velocity after 4.00 seconds?  (d) If the cord has stretched so that the carts are then 1.50 m apart, how far has each cart traveled during the 4.00 seconds? 

9.      Two rocks of mass 50.0 kg and 10.0 kg are floating in space, initially at rest relative to one another.  The two rocks are initially 100.0 m apart.  Due to universal gravitation the two rocks attract one another and accelerate until a collision occurs.  How much does the smaller rock move before the collision occurs?

10.  A gondolier of mass 75.0 kg is standing at one end of his symmetrical boat that is 6.0 m long and has a mass of 225 kg.  The boat is initially at rest.  The gondolier then walks to the other end at a speed of 1.5 m/s relative to the boat and stops.  Ignore friction or drag with the water.  (a) What is the velocity of the boat relative to shore while the gondolier is walking from one end to the other?  (b) How much has the boat moved by the time he reaches the other end?  (c) What is the final velocity of the boat?  (d) Which answers would be different is he moves faster from one end of the boat to the other?  Explain.  (e) How would the answers differ if drag is significant?  Explain.

11.  A 80.0 kg stuntman rides a 10.0 kg bicycle off a pier and into the water.  The pier is 4.00 m above the water and the initial speed of the bicycle is 12.0 m/s.  The stuntman jumps off the bicycle in midair and both the man and the bicycle hit the water at the same time.  If the bicycle is 8.00 m from the pier when it hits, where does the man hit the water?

12.  A rat of mass 150.0 g is at one end of a stationary skateboard with properties:  mass = 1.50 kg, length = 60.0 cm, μ = 0.100.  The rat scurries to the right across the top of the board to the other end and stops there.  This takes 1.50 s to occur.  (a) What will be the resulting velocity of the rat/skateboard system just as the rat stops?  (b) What is the displacement of the board during this interval? 

13.  A lab cart A of mass 1.50 kg traveling at 3.00 m/s, 0° collides with a cart B of mass 0.50 kg traveling at 1.00 m/s, 0°.  After the collision, cart A has velocity 2.00 m/s, 0°.  (a) Find the total momentum of the system.  (b) Find the amount of momentum transferred from one cart to the other.  (c) Find the speed of cart B after the collision.  (d) Repeat with everything the same except cart B is initially traveling at 180°. 

14.  For target practice, a bullet of mass 2.00 g is fired with a speed of 175 m/s into an old can of mass 25.0 g.  Find the speed of the can that will result from the impact of the bullet in each of the following scenarios:  (a) the bullet sticks in the side of the can, (b) the bullet bounces off the can and comes to a complete stop, (c) the bullet goes all the way through the can and leaves with speed 35 m/s.

15.  Find the recoil speed of a 1.70 kg rifle that fires a 12.0 g bullet at a speed of 485 m/s.

16.  A boy of mass 20.0 kg is rolling along in a 30.0 kg wagon at 3.00 m/s to the right over a level surface with little friction.  In the wagon is a brick of mass 1.50 kg.  With what velocity would the boy have to throw the brick in order to stop himself and the wagon? 

17.  Two amorphous cosmic blobs are drifting through intergalactic space.  Blob 1 has mass 45.0 kg and velocity 10.0 m/s, 90.0°, Blob 2 has mass 20.0 kg and speed 8.00 m/s, 180.0°.  The two blobs collide and stick together, becoming one amorphous space blob.  (a) Find the velocity of the resulting blob as it flies off through space.  (b) Find the total momentum of the system.  (c) Repeat for Blob 2 having an initial velocity of 44.0 m/s, 120.0°.

18.  A bullet of mass 10.0 g is fired into a ballistic pendulum of mass 3.00 kg.  The pendulum swings back and rises 12.0 cm.  Find the original speed of the bullet.

19.  A 15.0-g bullet with speed 650.0 m/s is fired into a ballistic pendulum of mass 2.00 kg.  How far does the pendulum rise?

20.  A spring is hung from the ceiling.  A pan of mass 100.0 g is attached to the end, which causes it to stretch 5.00 cm.  Find the maximum distance the pan moves downward when a lump of clay of mass 120.0 g is dropped from a height of 40.0 cm onto the pan.

21.  In classroom demonstration two lab carts one of mass 1.60 kg and the other 0.600 kg are set into motion by a spring of constant k = 75 N/m.  The spring is compressed by 3.00 cm between the two carts which are then released from rest.  If all of the energy stored in the spring is transferred to the carts, what is the resulting velocity of each cart?

22.  Imagine a boy sitting on a chair with his feet on a rung of the chair.  Could the boy move himself and the chair across the room without touching anything but the chair?  Discuss in terms of systems of particles, internal vs. external force, and conservation of momentum.

23.  When the brakes of a bicycle are applied, a small rubber pad rubs against the rim of the wheel.  (a) Is the action of the pad against the rim an internal or external force?  Justify your answer.  (b) If only an external force can affect the momentum of a system, what force stops the bicycle?  Explain.  (c) If momentum is always conserved what becomes of the bicycle’s momentum when it stops?  Explain.

24.  Analyze the scenarios in problem 13 and determine which is elastic and which is inelastic by finding the percent of kinetic energy that remains after the collision.

25.  A glider of mass 150.0 g  moves along an air track with velocity 3.00 m/s, 180.0° collides elastically with a second glider with mass 100.0 g and velocity 2.00 m/s, 0.0°.  Find the velocity of each glider on the track after the collision.

26.  A lump of clay, mass m, moving horizontally, collides with a block, mass M, which rests on a frictionless surface.  The clay sticks to the block.  (a) Show that the ratio of kinetic energy after to kinetic energy before the collision is:  m/(m + M).  (b) If energy is always conserved, what becomes of the missing kinetic energy?

27.  Two objects undergo a head-on elastic collision.  Object A has initial velocity 4 m/s, right and object B has initial velocity 3 m/s, left.  The result of the collision is that object B stops moving.  Determine the rebound velocity of object A and the ratio of the masses of the two objects.

28.  A proton of mass m moving with velocity 2.0 × 105 m/s, 0.0° collides elastically with an alpha particle of mass 4m that is at rest.  After the collision, the proton moves off in a direction of 30.0°.  (a) Find the speed of the proton after the collision.  (b) Find the velocity of the alpha after the collision.

29.  A hockey puck is moving with speed 20.0 m/s, 180.0° across frictionless ice.  It collides with an identical puck initially at rest.  After the collision one of the pucks is moving at 10.0 m/s.  Assume the collision is elastic.  (a) Find the resulting speed of the other puck.  (b) Find the direction of each puck’s movement after the collision.

30.  A golf ball of mass 45.9 g is at rest on a tee.  It is struck by a club of mass 225 g moving with velocity 30.0 m/s, 0.0°.  Assume the collision is perfectly elastic.  (a) Find the velocity of the center of mass.  (b) Find the velocity of each object relative to the center of mass before the collision.  (c) Find the velocity of each object relative to earth after the collision.

31.  A boy kicks a rolling ball of mass 405 g such that its velocity changes from 2.00 m/s, 0.0° to 15.0 m/s, 0.0°.  The boy’s foot exerts an average force of 575 N, 0.0° on the ball during the kick.  (a) Find the change in momentum of the ball.  (b) Find the amount of time the foot is in contact.

32.  A baseball of mass 145 g has velocity 35.0 m/s, 180.0° when it is hit by the bat.  The ball leaves the bat with a velocity of 50.0 m/s, 30.0° and sails over the pitcher’s head.  (a) Find the change in momentum of the ball.  (b) If the ball is in contact for 2.0 ms, what is the average force of the bat on the ball?

33.  The asteroid Apophis, m = 2.1 × 1010 kg, occupies an orbit that brings it periodically near the Earth.  Scientists speculate that one day it may be necessary to “nudge” it from its orbit in order to prevent a collision.  A change in velocity of 2.0 × 10−6 m/s may be sufficient if it occurs well in advance of the collision.  (a) Find the required amount of change in momentum.  (b) If an ion engine with thrust 0.090 N is used to do the nudging, how much time will be needed to deliver a sufficient impulse?

34.  Nasa’s Deep Space 1 spacecraft helped pioneer the technology of ion propulsion.  In its rocket engine ionized xenon atoms were accelerated by electric fields to speeds of 35 km/s and expelled through the nozzle into space.  (a) If the xenon fuel is used at a rate of 1.4 mg/s, what is the thrust of the engine?  (b) How much impulse could this engine deliver using the entire 81.5 kg of xenon fuel on board?

35.  Each solid rocket booster on the Space Shuttle produces 11.8 MN thrust and burns fuel at a rate of 3880 kg/s.  (a) Find the speed of the gases leaving the nozzle of the engine.  (b) The fuel is exhausted after 2.0 minutes; how much impulse has the booster delivered?

36.  The momentum of a certain object moving along the x-axis is given by p = 2t2 −5t + 3, where p is in kg m/s and t is in s.  (a) Find the net force on the object at t = 0.5 s.  (b)  Find the net force at the point in time when the object reverses its original direction of travel. (c) Find the net impulse on the object between t = 0 and t = 0.8 s.

37.  A ball of mass 0.550 kg dropped onto the floor and it bounces off.  The amount of force that the ball exerts on the floor can be modeled by:  F = at2 + bt.  Suppose a = −3.20 × 107 N/s2 and b = 3.50 × 105 N/s.  (a) Based on this model determine the amount of time that the ball is in contact with the floor.  (b) Find the maximum amount of normal force.  (c) Find the impulse of the ball acting on the floor.  (d) Find the impulse of gravity acting on the ball that occurs during the bounce.  (e) Find the change in momentum of the ball that occurs during the bounce.

38.  Consider a ball that bounces off of a surface at an angle.  Typically the path followed by the ball will make the same angle with the surface before and after the bounce.  Explain how this makes sense by referring to impulse, momentum, elasticity, etc.  Be specific.

39.  Nasa’s Dawn spacecraft has a total mass of 1210 kg, including 425 kg xenon fuel for its ion engine.  The ion engine uses fuel at a rate of 3.25 mg/s when operating at maximum thrust of 91.0 mN.  (a) Find the speed of the ions leaving the nozzle of the engine.  (b) Ignoring gravity, what total change in velocity could be achieved by running at full power until all of the fuel is used? (c) Find the minimum and maximum rates of acceleration that can occur with the engine at full thrust.

40.  A model rocket has a total mass of 190.0 g at liftoff (including fuel).  The rocket engine has an impulse rating of 17.0 Ns and burns 21.0 g of fuel in 1.60 s as the rocket shoots straight up.  Disregard air resistance.  (a) Find the speed of the rocket just as the engine cuts out (include the effect of gravity).  (b) Find the total time that the rocket will rise (before reaching maximum height).  (c) Sketch the speed vs. time graph for this interval.  (d) Estimate or explicitly determine the maximum height achieved.

 

 

1.      a. xCM = 3.8 m
b. x = −6.8 m

2.      4670 km from center of Earth

3.      1.26 m from big kid

4.      a. 1010 g
b. 1.32 m from the tail

5.       

6.       

7.     

8.      a. 0.400 m from large cart
b. 2.00 m/s2 in dir. of force
c. 16.0 m, 8.00 m/s in dir. of force
d. large cart:  15.8 m
    small cart:  16.3 m

9.      83.3 m

10.  a. 0.375 m/s opp. dir. of walk
b. 1.5 m opp. dir. of walk
c. 0
d.
e.

11.  11.2 m

12.  a. 1.47 m/s, right
b. 1.05 m, right

13.  a. 5.00 kg m/s, 0.0°
b. 1.50 kg m/s
c. 4.00 m/s
d. 4.00 kg m/s, 0.0°, 1.50 kg m/s,
    2.00 m/s

14.  a. 13.0 m/s
b. 14.0 m/s
c. 11.2 m/s

15.  3.42 m/s, opp. dir.

16.  100 m/s, right (unlikely!)

17.  a. 7.35 m/s, 109.6°
b. 478 kg m/s, 109.6°
c. 19.8 m/s, 110.0°,
    1290 kg m/s, 110.0°

18.  462 m/s

19.  1.19 m

20.  0.233 m

21.  smaller:  0.286 m/s
larger:  0.107 m/s, opp. dir.

22.   

23.  a.
b.
c.

24.  100%, elastic; 57%, inelastic

25.  larger:  1.00 m/s, 0.0°
smaller:  4.00 m/s, 180.0°

26.  a.
b.

27.  7 m/s, left; mA/mB = 3/11

28.  a. 1.93 × 105 m/s
b. 2.55 × 104 m/s, 288.6°

29.  a. 17.3 m/s
b. slower:  120.0°; faster: 210.0°
    (or 240.0° and 150.0°)

30.  a. 24.9 m/s, 0.0°
b. 5.08 m/s, 0.0° (club)
    24.9 m/s, 180.0° (ball)
c. 19.8 m/s, 0.0° (club)
    49.8 m/s, 0.0° (ball)

31.  a. 5.27 kg m/s, 0.0°
b. 0.00916 s

32.  a. 11.9 kg m/s, 17.7°
b. 5960 N, 17.7°

33.  a. 42000 kg m/s
b. 130 h

34.  a. 0.049 N
b. 2.9 × 106 Ns

35.  a. 3040 m/s
b. 1.4 × 109 Ns

36.  a. −3 N
b. −1 N
c. −2.7 Ns

37.  a. 0.0109 s
b. 957 N
c. 6.98 Ns, down
d. 0.0590 Ns, down
e. 6.92 kg m/s, up

38.   

39.  a. 28.0 km/s
b. 12.1 km/s
c. 7.52 × 10−5 m/s2 (fully laden)
    1.16 × 10−4 m/s2 (nearly empty)

40.  a. 79.1 m/s
b. 9.67 s
c.
d. 383 m (estimate), 381 m (explicit)