Resource Lesson
Springs and Blocks
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The following examples include an assortment of scenarios involving springs and blocks. Conservation of momentum and energy are common threads connecting them. Some also involve frictional forces and SHM.
We will begin with horizontal collisions, then progress to
vertical collisions
.
Colliding Blocks #1
Refer to the following information for the next two questions.
Two blocks I and II have masses
m
and
2m
respectively. Block II has an ideal massless spring (with force constant,
k
) attached to one side and is initially stationary while block I approaches it across a frictionless, horizontal surface with a speed
v
_{o}
. All momentum and energy initially belong to block I.
block I
mass = m
initial velocity v
_{o}
block II
mass = 2m
at rest
A. What would be the maximum compression of the spring during the collision?
B. Write an expression for the relative speed of block II after the blocks separate.
Colliding blocks #2:
Refer to the following information for the next three questions.
A massless spring with force constant k = 250 N/m is fastened at its left end to a vertical wall. Initially, block I (mass = 1.0 kg) and block II (mass = 2.0 kg) rest on a horizontal surface with block I in contact with the spring (but not compressing it) and with block II in contact with block I.
Block I is then moved to the left, compressing the spring a distance of 50 cm and held in place while block II remains at rest. (Use g = 10 m/s
^{2}
.)
A. How much elastic energy is stored in the compressed spring?
B. Block I is then released and accelerates to the right, toward block II. The surface is rough and the coefficient of friction between each block and the surface is µ = 0.25. The two blocks collide, stick together, and move to the right.
The speed
v
of block I just before it collides with block II is ____.
C. Remembering that the spring is not attached to block I, how far could the two blocks slide before coming to rest?
Stacked Blocks
Refer to the following information for the next four questions.
A small block (mass = 1 kg) rests on but is not attached to a larger block (mass = 2 kg) that slides on its base without friction. The coefficient of static friction between block I and block II is µ
_{s}
= 0.3 A spring with force constant k = 250 N/m attaches block II to the wall.
A. Determine the maximum horizontal acceleration that the 2-kg block may have without causing the 1-kg block to slip.
B. Determine the maximum amplitude A for simple harmonic motion of the two masses if they are to move together, i.e., the 1-kg block must not slip on the 2-kg block.
C. The two-mass combination is pulled to the right the maximum amplitude
A
found in part (b) and released. Describe the frictional force on the 1-kg mass during the first half cycle of oscillation.
D. The two-mass combination is now pulled to the right a distance of A' greater than A and released. Determine the accelerations of both the 1-kg block and the 2-kg block at the instant the masses are released.
Vertical Springs #1
Refer to the following information for the next four questions.
Suppose a 1-kg block is dropped from a height of 0.5 meter above an uncompressed spring. The spring has an elastic constant of 250 N/m and negligible mass. The block strikes the end of the spring and sticks to it. (Use g = 10 m/sec
^{2}
)
A. Determine the speed of the block at the instant it hits the end of the spring.
B. How far will the spring be compressed when the speed of the block is maximum?
C. As the spring oscillates in SHM, what is its amplitude?
D. What is the frequency of this spring/block system?
Vertical Springs #2
Refer to the following information for the next two questions.
A 2-kg block is fastened to a vertical spring that has a spring constant of 250 N/m. A 1-kg block rests on top of the 2-kg block. The blocks are now pushed down and released so that they oscillate.
A. How far can the spring be compressed and still allow the blocks to remain in contact at all times?
B. How fast will the blocks pass back through their original equilibrium position if they are released from the compressed distance calculated in Part A?
Incline Plane
Refer to the following information for the next two questions.
A "rocking box" apparatus can be used to determine coefficients of static and kinetic friction between a sliding block and the interior surface of the box. As the box is slowly rotated counterclockwise, the block of mass
m
will just start to slide when the box makes an angle
θ
with the horizontal. At this instant the box is stopped. Thus at angle
θ
, the block slides a distance
d
, hits the spring of force constant
k
, and compresses the spring a distance
x
before coming to rest.
A. Derive an expression for the coefficient of static friction, µ
_{s}
.
B. Derive an expression for the coefficient of kinetic friction, µ
_{k}
.
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