A block is attached to a horizontal spring; beginning with the spring compressed

Question

A block is attached to a horizontal spring; beginning with the spring compressed and the block at rest, the block is released into motion on a frictionless tabletop. The block has mass m = 6.68 kg, and the initial compression is 5.00 cm. After release, the spring begins to push the block back toward the equilibrium position at x = 0 cm. The graph shows the spring force on the block (in N) versus the position of the block (in cm) relative to equilibrium. Use the graph to answer the following questions. How much work is done by the spring in pushing the block from its initial position at x = -5.00 cm to x = 2.64 cm? What is the speed of the block when it reaches x = 2.64 cm? What is the maximum speed of the block?

Explanation / Answer

Here,

as F = k * x

k = F/x

from the graph ,

k = 6/.05 = 120 N/m

work done by the spring = 0.5 * k * (xi^2 - xf^2)

work done by the spring = 0.5 * 120 *(0.05^2 - 0.0264^2)

work done by the spring = 0.108 J

the work done by the spring is 0.108 J

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Using work energy theorum

0.5 * m * v^2 = work done by spring

0.5 * 6.68 * v^2 = 0.108

v = 0.18 m/s

the speed of the block when it reaches 2.64 cm is 0.18 m/s

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for the maximum speed ,

0.5 * m * v^2 = 0.5 * k * x^2

0.5 * 6.88 * v^2 = 0.5 * 120 * 0.05^2

v = 0.21 m/s

the maximum speed is 0.21 m/s

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