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Question 19.19: A trolley of mass m is fixed to the end of a spring. The spr...

A trolley of mass m is fixed to the end of a spring. The spring can be compressed and extended. The spring has a force constant k. The other end of the spring is attached to a vertical wall. The trolley lies on a smooth horizontal table. The trolley oscillates when it is displaced from its equilibrium position.
a Show that the motion of the oscillating trolley is s.h.m.
b Show that the period T of the trolley is given by the equation:

T = 2 \pi \sqrt{\frac{m}{k}}
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a The restoring force = kx (from Hooke’s law), a ∝ F, therefore a ∝ x. The force acts in the opposite direction to the displacement.

b a = – \frac{F}{m} = – \frac{kx}{m} = –  ω²  x

 

ω²  = \frac{k}{m}

 

ω = \sqrt{\frac{k}{m}}

 

f = \frac{ω}{2 \pi} = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}

 

T = \frac{1}{f} = 2 \pi \sqrt{\frac{m}{k}}

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