Thursday, 4 April 2013

Torque, Equilibrium & Oscillations

When an external force applied on a body has tendency to rotate about an axis, then the force is said to exert a torque on that body about the given axis. Let us consider a particle p whose position vector  and force  is acting on it. The torque on it is given by  =  x 

 = r F sin  

Magnitude = r F sin 

Force  can be resolved into its components along  (r) radial and perpendicular to  i.e., angular component 
  • Torque = Angular component x distance from the axis of rotation.
  • Torque = Force x perpendicular distance from axis of rotation.
  • There is no contribution to torque due to radial component of the force.
  • By right hand thumb rule, it can be seen that force figure (i), the cross product is anticlockwise and hence the torque is positive while force the second case it is negative.
  • If there are more than one force acting on a body, then net torque is given by
When a body is in a static equilibrium, it should neither accelerate nor begin to rotate due to force acting on it. i.e.,
  1. Net force net = 0
  2. Net torque net = 0
When two forces of equal magnitude and directed opposite to each other act on a body such that their line of action are separated by a distance, they form a couple of force.

Not torque due to the forces is along the axis of rotation in upward direction as the rotation is produced in anti-clockwise sense.

Not torque = Fx + Fy
= F (x + y) = Fd

The stability of a system is decided by the state of its equilibrium. There are three types of equilibrium.

Stable Equilibrium
Let us consider a pendulum. When it is displaced. When it is displaced from its equilibrium position an unbalanced force F termed as restoring force opposes the displacement i.e.  and  are anti-parallel.

Here  = -ve and F = 

U is minimum. For this

 > 0

i.e., potential energy is minimum at stable equilibrium.

Unstable Equilibrium

Let us consider a small block at the top of a smooth hemisphere. When it is slightly pushed, it slides downward i.e., it never returns to the equilibrium position,

The unbalanced force d and displacement  are in the same direction.

 is +ve where F = -and< 0. U is maximum at equilibrium position.

Neutral Equilibrium
A block on a table is slightly displaced from its position. It neither returns back nor moves from its initial position. i.e., the block does experience any net force.

Here = 0 and F = - = 0 and = 0

Simple Harmonic Motion
  1. Linear Simple Harmonic Motion
    In this case restoring force on the body of mass m is F = - k x

    So time period of oscillation of the body becomes
  2. Angular Simple Harmonic Motion
    In this case restoring torque acting on a body is  = - 0. So time period of oscillation of the body becomes
    where l = Motion of inertia of body about the point of oscillation.

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