PHY 53
Summer 2010
Assignment 7
Reading: Systems of Particles
Key concepts: Center of mass, momentum, motion relative to CM, collisions.
1.
Two identical balls of mass m are
propelled at t = 0 from the same
starting point. Ball A moves on a
frictionless horizontal surface, starting
with speed v0 . Ball B is thrown into
2v0
••
v0
the air with initial speed 2v0 , at elevation angle 60°.
2.
a.
What is the initial velocity (magnitude and direction) of the CM? Ans:
vCM (0) = ( 7 / 2) ⋅ v0 ; tan −1 ( 3 / 2) ≈ 41° above horizontal.
b.
What is the total external force on the system? Ans: mg .
c.
What is the acceleration (magnitude and direction) of the CM? Ans:
d.
Sketch on the drawing the trajectories of the balls and of the CM, while
ball B is in flight.
e.
Indicate on your sketch the location of the CM at the instant when ball B
has its maximum height.
1
g.
2
Some questions about frictionless ice.
a.
A pole is standing on end on frictionless ice as
shown. Its CM is at its midpoint. A slight nudge
makes it slip and fall. On the drawing sketch its
configuration when it is lying on the ice, and
explain your choice.
b.
Batman and Robin have been stranded by one of the arch-fiends in the
middle of a lake frozen with frictionless ice. Give a method by which they
can reach the shore.
c.
A child is standing at the front of a large raft resting on frictionless ice.
She walks to the back, then stops. Does the final location of the raft
depend on how fast she walks? Explain.
1
PHY 53
3.
Summer 2010
You are sitting in a cart at rest that can roll freely, facing the back. You have two
large bricks in the cart, each of mass m. The mass of you and the cart is M.
a.
You throw one brick off the back at speed v0 , and later throw the other off
at the same speed relative to the cart. What is the final speed of you and
the cart?
4.
b.
A second time, starting at rest, you throw both bricks off at the same time
with speed v0 . Now what is the final speed of you and the cart?
c.
Why does the second case result in a greater speed for you and the cart?
Three small balls are arranged as shown from above.
They are held in place on a frictionless talbe by pegs.
Between the top ball and the other two are two
identical springs, each compressed so that its potential
energy is U0 / 2 . The angle between the springs is 90°,
and the lower two balls are separated by distance R.
The springs are not attached to the balls.
2m
m
R
m
a.
Where is the CM of the system? [Use a coordinate system with origin
halfway between the lower balls, with y vertical and x horizontal.]
Indicate it on the drawing.
b.
The three pegs are simultaneously removed, allowing the balls to slide on
the table. Describe what happens. [What quantities are conserved?]
c.
Where is the CM of the system when the springs are no longer in contact
with the balls?
d.
In what direction does the ball of mass 2m move?
e.
What is its speed after the springs are no longer in contact with the balls?
2U0
[Give the answer in terms of m and U0 .] Ans:
.
3m
2
PHY 53
5.
Summer 2010
An unstable particle is moving along the x-axis with speed V when it decays into
two identical particles of mass m. The decay occurs at the origin of the lab frame.
In the reference frame moving with the original particle (which is the CM frame)
the decay gives total kinetic energy KCM to the resulting two particles. We are
interested in the velocities of the two particles in both frames.
Let the two particles move along the x-axis in the CM frame as shown.
y
CM frame
Lab frame
2
y
1
x
a.
x
Find the speeds of the two particles in the lab frame, in terms of m, V and
KCM . (Assume KCM > mV 2 .) Indicate their velocities in the lab frame with
arrows. Ans: v1 = v + V , v2 = v − V where v = KCM /m .
Now let the particles move along the y-axis in the CM frame, as shown.
CM frame
y
Lab frame
y
1
x
x
2
b.
For each, find the x and y components of velocity in the lab frame, in terms
of m, V and KCM . Ans: v1 = v2 = V , v1 = −v2 = v .
x
c.
x
y
y
For each, find the angle the velocity vector makes with the x-axis in the lab
frame. Indicate their velocities in the lab frame with arrows.
3
PHY 53
6*.
Summer 2010
A proton of mass m and speed v0 collides elastically with a deuteron of mass 2m
initially at rest. The final speed of the proton is twice that of the deuteron.
a.
Let the incoming proton move along the x-axis. Draw a picture showing
the situation after the collision, with the proton’s final velocity making
angle θ with the x-axis and the deuteron’s final velocity making angle φ
with that axis.
8.
b.
Find the final speeds of the two particles. Ans: vd = v0 / 6 , v p = 2v0 / 6 .
c.
Find the two angles. [The answer is numerical.] Ans: Both are
cos−1 ( 6 / 4) .
A pair of masses are connected by a massless rigid rod
of length  as shown. This system is to be rotated
about a fixed axis perpendicular to the rod.
a.
2m
If the axis passes through the CM, what is the
moment of inertia? Ans:
9.
Axis
2
m 2 .
3
b.
Let the axis pass through the geometric center
of the rod. Find the moment of inertia by direct
calculation. Ans: 34 m 2 .
c.
Verify that your answers satisfy the parallel axis theorem.
m

Questions about moments of inertia of solids.
a.
You have a solid sphere, a solid circular cylinder, and a hoop. All have
mass M and radius R. Rank them in order of increasing I about their axis
of symmetry, and explain the reasons for your choices.
b.
The moment of inertia of a thin uniform rod about an axis perpendicular
1
to it and through its center is I = 12
M 2 , where M is the mass and  is the
length. Show that for an axis perpendicular to the rod but through one
end I = 13 M 2 . [Do not do an integral.]
c.
The brick shown has three symmetry axes, each passing
through the center of one side, perpendicular to it. Rank the
moments of inertia of the axes passing through each side, and
give your reasons.
4
c
a
b