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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Simple Harmonic Motion Know More

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Damped Simple Harmonic Motion:

We know that a simple pendulum does not continue to oscillate endlessly. The amplitude of oscillation keeps decreasing and after some time the pendulum stops oscillating. If the pendulum is placed under water, it will swing only briefly. This happens because, in the first case, air, and in the second, water, exert a drag force on the pendulum and transfer energy from the motion of the pendulum.
When the motion of an oscillator is reduced due to some external force, we say the motion is 'damped'.

Consider a block of mass m attached to a spring with spring constant k and oscillating vertically. Let a vane connected by a rod to the block (both assumed to be massless) be immersed in a liquid as shown. As the mass oscillates, the liquid will exert a drag force on the vane and thus on the entire system. The mechanical energy of the block-vane system is gradually transformed to thermal energy of the liquid and the vane.

The damping force Fd is proportional to the velocity of the vane.
Thus Fd =
-bv where b is a constant. The negative sign indicates that Fd opposes the motion.
The force F
s on the block due to the spring is given by Fs = -kx
Then,
ma = -bv - kx (Newton's 2nd law)

i.e., m.

d2x

= -

dx

 - kx

dt2

dt

 

or m .

d2x

 +

dx

 + kx = 0

dt2

dt


The solution of this equation gives
x = A e-bt/2 m cos (w't + f), [where A is the amplitude and w' is the angular frequency of the damped oscillator.]
w' =
If there is no damping,
b = 0 and w = , which gives the angular frequency of an undamped oscillator.

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