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A spherical bob is chosen for a simple pendulum for the following reasons :

1. For a given value of density, the spherical shape has minimum volume.
2. The air friction is minimum due to spherical shape.

For a simple pendulum
E_total = \(\frac { 1 }{ 2 } \) mω² a² = \(\frac { 1 }{ 2 } \)m(2πn)² a²
= \(\frac { 1 }{ 2 } \) m 4π² n² a²
= 2π² mn² a²
Here, E_total = total energy
m = mass of the pendulum (bob)
ω = angular frequency = 2πn = 2π/T
n = linear frequency
a = amplitude.

The phase difference between displacement and acceleration is π or 180°.

The time taken by the body starting its simple harmonic motion from the extreme end to the mean position is T/4 and accordingly the phase angle is π/2. Hence, the initial phase angle is π/2.

Total mechanical energy is conserved in simple harmonic motion.

In a simple harmonic motion, the displacement and acceleration are in the opposite direction, displacement is ahead of acceleration by a phase angle π.

For a simple harmonic motion, acceleration is directly proportional to the displacement of the object from its mean position and always points towards the mean position (f = -ω²y).

The relation between the time period and the spring constant of an oscillating body is

\(T = 2\pi \sqrt{\frac{m}{k}}\)

k = spring constant
The spring constant for a softer spring is less than that for a harder one.
i.e., K_soft < K_hard
Hence from the above equation
T_soft > T_hard
i. e, the time period will increase.

Formula for the time period of a simple pendulum,

\(T = 2 \pi \sqrt{\frac{l}{g}}\)

In this expression, there is no term ‘a’, hence the time period is independent of the amplitude of oscillation. Thus, the time period remains unaffected.

If there is only restoring force acting on a vibrating object for example simple pendulum or tuning fork then that object can vibrate till infinity. This type of vibration or oscillation is called ‘free oscillation’ and this type of vibration is called ‘free vibration’. Its frequency is called ‘Natural Frequency’. There is only restoring force working in free vibration. Its amplitude and energy both are constant with time.