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Oscillations due to a Spring

Let us consider a block of mass which is attached to a spring, which in turn is fixed to a rigid wall. The block is placed on a frictionless horizontal surface. If the block is pulled on one side and is released, it then executes a to and fro motion about a mean position.

According to Hooke’s law, if a system is deformed, it is subject to a restoring force, the magnitude of which is proportional to the deformation or the displacement and acts in opposite direction.
F(x) = - K(x)

where k is the spring constant. We can relate
ω and k by the given relation,

and hence the time period can be written as,

Simple Harmonic Oscillator

A simple pendulum consists of a mass m hanging from a string of length and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion.

By applying Newton's second law for rotational systems, the equation of motion for the pendulum may be obtained


Sin θ can be expanded as

where θ is in radians. For small angle Sin θ can be approximated as θ . Hence the above equation can be written as,

If the amplitude of angular displacement is small enough that the small angle approximation (Sin θ θ ) holds true, then the equation of motion reduces to the equation of simple harmonic motion with the angular frequency.
and the time period of the pendulum is

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