C1989M3. A 2-kilogram block is dropped from a height of 0.45 meter above an uncompressed spring, as shown above. The spring
has an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it.
a. Determine the speed of the block at the instant it hits the end of the spring
b. Determine the force in the spring when the block reaches the equilibrium position
c. Determine the distance that the spring is compressed at the equilibrium position
d.
Determine the speed of the block at the equilibrium position
e.
Determine the resulting amplitude of the oscillation that ensues
f.
Is the speed of the block a maximum at the equilibrium position, explain.
g.
Determine the period of the simple harmonic motion that ensues
1990M3. A 5-kilogram block is fastened to an ideal vertical spring that has an unknown
spring constant. A 3-kilogram block rests on top of the 5-kilogram block, as shown above.
a. When the blocks are at rest, the spring is compressed to its equilibrium position a
distance of ∆x1 = 20 cm, from its original length. Determine the spring constant of
the spring
The 3 kg block is then raised 50 cm above the 5 kg block and dropped onto it.
b. Determine the speed of the combined blocks after the collision
c. Setup, plug in known values, but do not solve an equation to determine the amplitude ∆x 2 of the resulting oscillation
d.
Determine the resulting frequency of this oscillation.
e.
Where will the block attain its maximum speed, explain.
f.
Is this motion simple harmonic.
(2000 M1) You are conducting an experiment to measure the acceleration due to gravity g u at an unknown location. In the
measurement apparatus, a simple pendulum swings past a photogate located at the pendulum's lowest point, which records the time t10
for the pendulum to undergo 10 full oscillations. The pendulum consists of a sphere of mass m at the end of a string and has a length
l. There are four versions of this apparatus, each with a different length. All four are at the unknown location, and the data shown
below are sent to you during the experiment.

a.
b.
(cm)
12
t10
(s)
7.62
18
8.89
21
10.09
32
12.08
T
(s)
T2
(s2)
For each pendulum, calculate the period T and the square of the period. Use a reasonable number of significant figures. Enter
these results in the table above.
On the axes below, plot the square of the period versus the length of the pendulum. Draw a best-fit straight line for this data.
c.
Assuming that each pendulum undergoes small amplitude oscillations, from your fit, determine the experimental value gexp of the
acceleration due to gravity at this unknown location. Justify your answer.
d.
If the measurement apparatus allows a determination of gu that is accurate to within 4%, is your experimental value in agreement
with the value 9.80 m/s2 ? Justify your answer.
e.
Someone informs you that the experimental apparatus is in fact near Earth's surface, but that the experiment has been conducted
inside an elevator with a constant acceleration a. If the elevator is moving down, determine the direction of the elevator's
acceleration, justify your answer.
C2003M2.
An ideal massless spring is hung from the ceiling and a pan suspension of total mass M is
suspended from the end of the spring. A piece of clay, also of mass M, is then dropped from a
height H onto the pan and sticks to it. Express all algebraic answers in terms of the given
quantities and fundamental constants.
(a) Determine the speed of the clay at the instant it hits the pan.
(b) Determine the speed of the clay and pan just after the clay strikes it.
(c) After the collision, the apparatus comes to rest at a distance H/2 below the current position. Determine the
the attached spring.
(d) Determine the resulting period of oscillation
spring constant of
C2008M3
In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs
the cord from a rod as represented above and then attaches a variety of weights to the cord. For each
weight, the student allows the weight to hang in equilibrium and then measures the entire length of
the cord. The data are recorded in the table below:
(a) Use the data to plot a graph of weight versus length on the axes below. Sketch a best-fit straight line through
the data.
(b) Use the best-fit line you sketched in part (a) to determine an experimental value for the spring constant k of the cord.
The student now attaches an object of unknown mass m to the cord and holds the object adjacent to
the point at which the top of the cord is tied to the rod, as shown. When the object is released from
rest, it falls 1.5 m before stopping and turning around. Assume that air resistance is negligible.
(c) Calculate the value of the unknown mass m of the object.
(d) i. Determine the magnitude of the force in the cord when the when the mass reaches the
equilibrium position.
ii. Determine the amount the cord has stretched when the mass reaches the equilibrium position.
iii. Calculate the speed of the object at the equilibrium position
iv. Is the speed in part iii above the maximum speed, explain your answer.
Supplemental
One end of a spring of spring constant k is attached to a wall, and the other end is attached to a block of mass M, as shown above. The
block is pulled to the right, stretching the spring from its equilibrium position, and is then held in place by a taut cord, the other end of
which is attached to the opposite wall. The spring and the cord have negligible mass, and the tension in the cord is F T . Friction
between the block and the surface is negligible. Express all algebraic answers in terms of M, k, F T , and fundamental constants.
(a) On the dot below that represents the block, draw and label a free-body diagram for the block.
(b) Calculate the distance that the spring has been stretched from its equilibrium position.
The cord suddenly breaks so that the block initially moves to the left and then oscillates back and forth.
(c) Calculate the speed of the block when it has moved half the distance from its release point to its equilibrium position.
(e) Calculate the time after the cord breaks until the block first reaches its position furthest to the left.
(e) Suppose instead that friction is not negligible and that the coefficient of kinetic friction between the block and the surface is µk .
After the cord breaks, the block again initially moves to the left. Calculate the initial acceleration of the block just after the cord
breaks.