Question

Derive the relation between torque and moment of inertia?

Answer 1

Hint: TORQUE- Torque is the measure of the force that can cause an object to rotate about an axis. Force is what causes an object to accelerate in linear kinematics, similarly, torque is what causes an angular acceleration. Hence, torque can be defined as the rotational equivalent of linear force.
MOMENT OF INERTIA- The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

Step-By-Step answer:
Now we will find the relation between torque and moment of inertia-
When a torque acts on a body rotating about an axis, it produces an angular acceleration in the body.
Let, the angular velocity of each particle be $\omega$. Then,
Angular acceleration $(\alpha ) = \dfrac{{d\omega }}{{dt}}$
The linear acceleration will depend on their distance ${r_1},{r_2}......{r_n}$ from the axis of rotation.
Consider a particle P of mass ${m_1}$ at a distance ${r_1}$. Let its linear velocity be ${v_1}$.
Linear acceleration of 1st particle = ${a_1} = {r_1}\alpha$
Force acting on 1st particle = ${F_1} = {m_1}{r_1}\alpha$
Moment of force ${F_1}$ about axis of rotation is
${\tau _1} = {F_1}{r_1} = {m_1}{r_1}^2\alpha$
Total torque = $\tau = {\tau _1} + {\tau _2} + .......... + {\tau _n}$
$\tau = {m_1}{r_1}^2\alpha + {m_2}{r_2}^2\alpha + ....... + {m_1}{r_n}^2\alpha$
$\tau = ({m_1}{r_1}^2 + {m_2}{r_2}^2 + {m_3}{r_3}^2 + ....... + {m_n}{r_n}^2)\alpha$
$\tau = (\sum\limits_{}^{} {m{r^2}} )\alpha$
$\tau = I\alpha$, where I is the moment of inertia.
This is the relation between torque and moment of inertia.

NOTE- The torque produced in a body makes the body rotate about an axis, which is called the axis of rotation. In physics, torque is simply the tendency of a force to turn or twist. The formula used to calculate the torque is given by $\tau = F.r.\sin \theta$. Unit of torque is Newton-meter (N-m).