- Periodic Motion
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- Simple Harmonic Motion
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- Equation of SHM
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- Characterstics of SHM
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- Velocity of SHM
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- Acceleration of SHM
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- Total Energy of SHM
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- Motion of a body suspended from a spring
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- Simple pendulum
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- The compound pendulum
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- Damped Oscillations
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- Driven or Forced Harmonic oscillator

- Simple pendulum consists of a point mass suspended by inextensible weightless string in a uniform gravitational field.
- Simple pendulum can be set into oscillatory motion by pulling it to one side of equilibrium position and then releasing it.
- In case of simple pendulum path ot the bob is an arc of a circle of radius l, where l is the length of the string.
- We know that for SHM F=-kx and here x is the distance measured along the arc as shown in the figure below.

- When bob of the simple pendulum is displaced from its equilibrium position O and is then released it begins to oscillate.
- Suppose it is at P at any instant of time during oscillations and θ be the angle subtended by the string with the vertical.
- mg is the force acting on the bob at point P in vertically downward direction.
- Its component mgcosθ is balanced by the tension in the string and its tangential component mgsinθ directs in the direction opposite to increasing θ .
- Thus restoring force is given by

F=-mgsinθ (19) - The restoring force is proportional to sinθ not to The restoring force is proportional to sinθ, so equation 19 does not represent SHM.
- If the angle θ is small such that sinθ very narly equals θ then above equation 19 becomes

F=-mgθ

since x=lθ then,

F=-(mgx)/l

where x is the displacement OP along the arc. Thus,

F=-(mg/l)x (20) - From above equation 20 we see that restoring forcr is proportional to coordinate for small displacement x , and the constant (mg/l) is the force constant k.
- Time period of a simple pendulum for small amplitudes is

- Corresponding frequency relations are

and angular frequency

ω=√(g/l) (23) - Notice that the period of oscillations is independent of the mass m of the pendulumand for small oscillations pperiod of pendulum for given value of g is entirely determined by its length.

- Compound pendulum is a rigid body of any shape, capable of oscillating about a horizontal axis passing through it.
- Figure below shows vertical section of rigid body capable of oscillating about the point A.

- Distance l between point A and the centre of gravity G is called length of the pendulum.
- When this compound pendulum is given a small angulr displacement θ and is then released it begins to oscillate about point A.
- At angular displacement θ its center of gravity now takes new position at G'.
- Weight of the body and its reaction at the support constitute a reactive couple or torque given by

τ=-mg G'B

=-mglsinθ (24) - Equation 24 gives restoring couple which tends to bring displaced body to its original position.
- If α is the angular acceleration produced in this body by the couple and I is the moment of inertia of body about horizontal
axis through A then the couple is

Iα=-mglsinθ

if θ is very small then we can replace sinθ≅θ, so that

α=-(mgl/I)θ (25) - From above equation (25) we se that pendulum is executing Simple Harmonic Motion with time period

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