derive a relation between torque and angular momentum
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It plays the same role in rotational dynamics as force plays in linear motion. Torque is defined as the moment of force or turning effect of force about the given axis or point. It is measured as the cross product of position and force vector while as angular momentum is the rotational analogue of linear momentum.
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Angular momentum(L):- It is an measure of the rotation of body, taking into account its mass , shape and speed.
L = r × p
where r is distance from the axis of rotation and the p is the linear momntum of the body.
Torque(τ): It is an moment of force i.e. tendency of a force to rotate an object about an axis.
We have shown that Newton's second law can be written as
ΣF = dp/dt.
where ΣF is the net force acting on a body. Thus we can write
r ´ ΣF = r ´ dp/dt.
L = r × p
where r is distance from the axis of rotation and the p is the linear momntum of the body.
Torque(τ): It is an moment of force i.e. tendency of a force to rotate an object about an axis.
We have shown that Newton's second law can be written as
ΣF = dp/dt.
where ΣF is the net force acting on a body. Thus we can write
r ´ ΣF = r ´ dp/dt.
Since the net torque acting on a body is Στ = r ´ ΣF, and since we can add to the right hand side of the preceding equation the quantity dr/dt ´ p = 0 without changing its value, we can write
Στ = r ´ dp/dt + dr/dt ´ p = d(r ´ p)/dt.
We define angular momentum as
L º r ´ p
The preceding equation then becomes
Στ = d L/dt,
Στ = r ´ dp/dt + dr/dt ´ p = d(r ´ p)/dt.
We define angular momentum as
L º r ´ p
The preceding equation then becomes
Στ = d L/dt,
which is the rotational analogue of Newton's second law as expressed at the top of this derivation. (We note that
like torque, which depends both upon the net force applied to a body and the point about which that torque is
calculated, a body's angular momentum depends upon the net momentum of the body and upon the point about
which its angular momentum is calculated.)
like torque, which depends both upon the net force applied to a body and the point about which that torque is
calculated, a body's angular momentum depends upon the net momentum of the body and upon the point about
which its angular momentum is calculated.)
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