A block of mass m is attached to a horizontal spring of constant k on a frictionless…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A block of mass m is attached to a horizontal spring of constant k on a frictionless surface. Initially the block is released from rest at amplitude

A_0

relative to the equilibrium position. Just as the block passes through equilibrium for the first time, an external agent gives it an instantaneous additional speed

v_p

in the same direction as its motion and then removes the push. The spring constant remains unchanged, and the block continues oscillating. (a) Write an expression for the total mechanical energy of the system before the push in terms of k and

A_0

. (b) Write an expression for the total mechanical energy immediately after the push in terms of k,

A_0

v_p

, and m. (c) Derive an algebraic expression for the new amplitude

A_{\text{new}}

that results from the added speed. Express

A_{\text{new}}

in terms of

A_0

, m, k, and

v_p

. (d) Evaluate

A_{\text{new}}

for

m = 0.40\,\text{kg}

k = 120\,\text{N\,m}^{-1}

A_0 = 0.10\,\text{m}

, and

v_p = 0.60\,\text{m\,s}^{-1}

. Show all steps and include units.

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