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Tuesday, 28 February 2017

Law of Motion XIX


Tension at lowermost point in critical condition 
For lowermost point B applying net force towards center is equal to centripetal force. Tension Tacts towards the center of the circular path whereas weight mg acts away from it in vertically downward direction. Hence,
TB – mg = mvB 2 / r
Putting, vB = √4rg
TB – mg = m4gr/r

or, TB = 5mg
Hence in critical condition of vertical circular motion of a body attached to the rod velocities at topmost and lowermost be 0 and √4rg respectively and tensions in the rod be mg (pushing nature) and 5mg (pulling nature) respectively.

Motion of A Body Over Spherical Surface 

A body of mass m is moving over the surface of the smooth sphere of radius r. At any instant when the radius of sphere passing through the body makes angle θ with the vertical the tangential velocity of the body is v. Since net force towards the center is centripetal force we have

mgCosθ – N = mv2 / r

or, N = mgCosθ – mv2 /r

if v increases N decreases and when the body just loses contact with the sphere N = 0

Putting N = 0,
0 = mgCosθ – mv2 / r
or, mv2 / r = mgCosθ

or, v = √rg Cosθ

This is the minimum velocity at which the body loses contact and it is the maximum velocity at which the body remains in contact with the surface

CENTRIFUGAL FORCE
It is a pseudo force experienced by a body which is a part of the circular motion. It is a non-realistic force and comes into action only when the body is in a circular motion. Once the circular motion of the body stops, this force ceases to act. Its magnitude is exactly same as that of centripetal force but it acts opposite to the direction of the centripetal force that is in the radially outward direction.
Frame of reference attached to a body moving on a circular path is a non-inertial frame since it an accelerated frame. So when ever any body is observed from this frame a pseudo force F = ma = mv2 /r = mrωmust be applied on the body opposite to the direction of acceleration along with the other forces. Since the acceleration of the frame in circular motion is centripetal acceleration a = v2/r directed towards the center of the circular path, the pseudo force applied on the bodies observed from this frame is F = mv2 /r directed away from the center of the circular path. This pseudo force is termed as a centrifugal force

CENTRIFUGE
It is an apparatus used to separate cream from milk. It works on the principal of centrifugal force. It is a cylindrical vessel rotating with high angular velocity about its central axis. When this vessel contains milk and rotates with high angular velocity all the particles of milk start moving with the same angular velocity and start experiencing centrifugal force FCentrifugal = mrωin radially outward direction. Since centrifugal force is directly proportional to the mass of the particles, massive particles of milk on experiencing greater centrifugal force starts depositing on the outer edge of the vessel and lighter cream particles on experiencing smaller centrifugal force are collected near the axis from where they are separated apart.


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