Establish the relation between Torque and Moment of Inertia (in rotational motion, between torque and angular acceleration).

Question

Accepted Answer

>

> [!TLDR]
> Summing $	au_i = m_ir_i^2\alpha$ over all particles of a rigid body gives the rotational analogue of Newton's second law, $	au=I\alpha$.
>
> Consider a rigid body rotating about a fixed axis with angular acceleration $\alpha$. Take a small mass element $m_i$ at perpendicular distance $r_i$ from the axis. Its tangential (linear) acceleration is $a_i = r_i\alpha$, so by Newton's second law the tangential force needed on it is $F_i = m_i a_i = m_i r_i\alpha$.
>
> The torque produced by this force about the axis is:
> $$	au_i = F_i\, r_i = m_i r_i^2 \alpha$$
>
> Summing over all mass elements of the rigid body (all share the same $\alpha$ since the body is rigid):
> $$	au = \sum_i 	au_i = \Big(\sum_i m_i r_i^2\Big)\alpha$$
> …

Linear Quantity	Rotational Analogue
Displacement $x$	Angular displacement $\theta$
Velocity $v$	Angular velocity $\omega$
Acceleration $a$	Angular acceleration $\alpha$
Mass $m$	Moment of inertia $I$
Force $F$	Torque $\tau$
Newton's 2nd law: $F = ma$	$\tau = I\alpha$
Kinetic energy: $\frac{1}{2}mv^2$	$\frac{1}{2}I\omega^2$
Momentum: $p = mv$	Angular momentum: $L = I\omega$

Q.Establish the relation between Torque and Moment of Inertia (in rotational motion, between torque and angular acceleration).

Concept understanding — Rotational Dynamics

Rotational Dynamics: The Physics of Spinning Things

The Core Idea

Breaking It Down

Torque — The Rotational "Push"

Moment of Inertia — The Rotational "Mass"

Angular Acceleration — How Fast Rotation Changes

The Complete Picture: Rotational Analogues

The Key Insight: Conservation of Angular Momentum

Q.Establish the relation between Torque and Moment of Inertia (in rotational motion, between torque and angular acceleration).

🎓Unlock everything free for 14 days

Unlock everything free for 14 days