Question

derive the mathematical relation between angular momentum and torque

Answer 1

Angular momentum L
L :  It is an measure of the rotation of body, taking into account its mass,             shape and speed.

L = r ×â p
where r is distance from the axis of rotation and the p is the linear momntum of the body.


Torque τ
τ: It is an moment of force i.e. tendency of a force to rotate an object about an axis.

τ = r × f
where, r is the perpendicular distance from the axis of the rotation and f is the applied force.


For rotational motion torque is given by:
τ = Iα ------------------------------------------- 1

where I is the moment of inertia of the body about an axis of rotation.


α is the angular acceleration.
We know for the rotational body angular momentum is conseverd.
L = Iω -------------------------------------------- 2


where, ω is the angular velocity.



As we know
α =  dω/dt ---------------------------------------3


taking derivative of equqtion no. 2,  with respect to time, we get.
dL/dt = dω/dt -----------------------------------4


Using
3 &4

 we get.
dL/dt = Iα --------------------------------------- 5


compare equation no 1 & 5

we get
ττ= dL/dt

This is the relationship between torque and angular momentum.


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