One end of a spring is attached to a solid wall while the other end just reaches
ID: 1910009 • Letter: O
Question
One end of a spring is attached to a solid wall while the other end just reaches to the edge of a horizontal, frictionless tabletop, which is a distancehabove the floor. A block of massMis placed against the end of the spring and pushed toward the wall until the spring has been compressed a distancex, as shown above. The block is released, follows the trajectory shown, and strikes the floor a horizontal distanceDfrom the edge of the table. Air resistance is negligible.
Determine expressions for the following quantities in terms ofM,x,D,h, andg. Note that these symbols do not include the spring constant.
(a) The time elapsed from the instant the block leaves the table to the instant it strikes the floor
(b) The horizontal component of the velocity of the block just before it hits the floor
(c) The work done on the block by the spring
(d) The spring constant
Explanation / Answer
a)
The time elapsed from the instant the block leaves the table to the instant it strikes thefloor.
This time is controlled by the time it takes for the block to fall.
y = 1/2at2
t = (2y/a)
t = (2h/g)
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b)
The horizontal component of the velocity of the block just before it hits the floor.
Velocity is constant in the x direction. We’ve figured out the time of flight.
v = x/t
= D / (2h/g)
= D (g/2h)
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c)
The work done on the block by the spring.
Let’s use conservation of energy to solve this one. (Finally we get into work and energy.)
W = 1/2 mv2
= 1/2 M [D (g/2h)]2
= MD2g / 4h
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d)
The spring constant.
The work is also equal to the potential energy of the spring.
1/2 kx2 = W
1/2 kx2 = MD2g / 4h
k = MD2g / 2hX2
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