A block of mass m = 5 .78 kq is attached to a spring which is resting on a horiz

Question

A block of mass m = 5 .78 kq is attached to a spring which is resting on a horizontal frictionless table. The block is pushed into the spring, compressing it by 5.00 m, and is then released from rest. The spring begins to push the block back toward the equilibrium position at x = 0 m. The graph shows the component of the force (in N) exerted by the spring on the block versus the position of the block (in m) relative to equilibrium. Use the graph to answer the following questions. How much work is done by me spring in pushing the block from its initial position at x = -5.00 m to x = 2.86 m? What is the speed of the block when it reaches x = 2.86 m? What is the maximum speed of the block?

Explanation / Answer

Here ,

let the spring force constant is k

force constant , k = F/x

this is the slope of graph

k = 6/5 N/m

k = 1.2 N/m

Now , work done by spring = - change in potential energy of spring

work done by spring = - ( 0.5 * 1.2 * 2.86^2 - 0.5 * 1.2 * 5^2)

work done by spring = 2.59 J

the work done by spring is 2.59 J

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Using work energy theorum

kinetic energy of block = work done by spring force

0.5 * 5.78 * v^2 = 2.59

v = 0.95 m/s

the speed of the block is 0.95 m/s

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for the maximum speed of the block is v

Using consevration of energy

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

5.78 * v^2 = 1.2 * 5^2

v = 2.55 m/s

the maximum speed of the block is 2.55 m/s

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