An ideal spring is lying horizontally on a frictionless surface.One end of the s

Question

An ideal spring is lying horizontally on a frictionless surface.One end of the spring is attached to a
wall. The other end is attached to a moveable block that has a massof 5 kg. The block is pulled so
that the spring stretches from its equilibrium position by 0.65 m.Then the block is released (from
rest), and as a result the system oscillates with a frequency of0.40 Hz (that's 0.40 rev/sec).
Find: (a) the acceleration of the block when the spring isstretched by 0.28 m.
(b) the maximum force magnitude exerted by the spring on theblock.
(c) the oscillation frequency of a 2.5 kg block under the samecircumstances (same spring
and initial displacement.

Please show all steps.

Explanation / Answer

Let the spring constant is k N/m
=>By T = 2pi x sqrt[m/k]
=>1/n = 2pi x sqrt[m/k]
=>1/0.40 = 2 x 3.14 x sqrt[5/k]
=>sqrt[5/k] = 0.40
=>[5/k] = (0.40)^2
=>k = 5/0.16
=>k = 31.55 N/m
(a) By m x d^2x/dt^2 = -kx
=>5 x (a) = -31.55 x 0.28
=>a = -1.77 m/s^2 [-ve indicating the direction only]
(b) F = -kx
=>F(max) = -31.55 x 0.65 = -20.51 N [-ve indicating the direction only]
(c) By T = 2pi x sqrt[m/k]
=>1/n = 2 x 3.14 x sqrt[2.5/31.55]
=>1/n = 1.77
=>n = 1/1.77 = 0.57 Hz

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