Problem 1 (25 pts)
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latexit
|~F1| = F
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latexit
|~F2| = 2F
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latexit
|~F3| = F
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latexit
60�
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latexit
30�
Three forces ~F1, ~F2 and ~F3 applies on an object of mass m. The direction
of the three forces is indicated in the right figure. The magnitude of
the forces ~F1 and ~F3 is F , and the magnitude of ~F2 is 2F . What is the
acceleration of the object? Express it in terms of F and m and remembe
that an acceleration is a vector.
2
Problem 2 (25 pts)
Surface
Three blocks of masses mA, mB and mC respectively are
stacked on top of each other as shown in the right figure.
A force ~F is applied on the middle block B. Suppose the
coefficient of static friction between the blocks is µs and
the coefficient of kinetic friction between block C and the
surface is µk. In below, express your answers in terms of
mA, mB, mC , ~F , µs, µk and/or g. (You may not need all
the variables.)
(a) (6 pts) If all the three blocks move with a constant
speed v when the force ~F is applied, what is the
frictional force between block A and B?
(b) (6 pts) Continue from part (a), what is the frictional force between the surface and block C?
Now, suppose the surface is frictionless, and the three blocks still move together.
(c) (6 pts) What is the acceleration of the three blocks?
(d) (7 pts) What is the frictional force between block B and C?
3
Problem 3 (25 pts) A
BConsider Earth o
iting around the Sun along an elliptical o
it. The
mass of the Sun and Earth are M and m respectively. When Earth passes
through point A in the o
it, it has a distance r to the Sun and a speed
v. When Earth passes through point B, it has a distance
3 to the Sun.
(a) (12 pts) What is the moving speed of Earth when it passes through point B?
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latexit>m
(b) (13 pts) This part is independent from part (a). Three springs of
spring constant k, 2k and 3k are connected with each other as shown in
the right figure. Namely the spring k and 2k are both attached to the
wall and connected to a movable plate who mass is negligible. The plate
is then connected to the spring 3k, which is attached to a block of mass m.
Find the effective spring constant keff of the spring system. That
is, if the block is displaced from its equili
ium position by a displacement
x, then the spring force acting on the block is F = −keff x.
4
Problem 4 (25 pts)
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latexit
✓
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latexit>v0
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latexit
R
A small bead of mass m slides down from the top of an hemisphere with an
initial speed v0 in the horizontal direction. The speed v0 is small enough
so that the bead does not leave the surface at the top. The radius of the
hemisphere is R and we assume the surface between the bead and the
hemisphere is frictionless. The hemisphere is held fixed throughout the
whole process.
(a) (13 pts) Suppose the bead still remain in contact with the hemi-
sphere when it is at an angle θ with respect to the vertical as shown
in the right figure. What is the bead’s speed at that point? Express
your answer in terms of v0, R, θ and g.
(b) (12 pts) At what height h will the bead leave the surface? Express it in terms of v0, R and g.
5