What do you mean by torque? Derive an expression for torque in terms of moment of inertia of a body about a given axis of rotation.

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Accepted Answer

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> [!TLDR]
> Summing $	au_i = m_i r_i^{2}\alpha$ over all particles gives $	au = \left(\sum m_i r_i^{2}ight)\alpha = I\alpha$.
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> **Torque** is the turning (rotational) effect of a force about an axis, defined as $\vec	au = \vec r	imes\vec F$; its magnitude is $	au = F\,r\sin	heta = F	imes(	ext{perpendicular distance})$.
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> **Derivation of $	au = I\alpha$:** Consider a rigid body rotating about a fixed axis with angular acceleration $\alpha$. Take a particle of mass $m_i$ at perpendicular distance $r_i$ from the axis. Its linear (tangential) acceleration is $a_i = r_i\alpha$, so the force needed is
> $$F_i = m_i a_i = m_i r_i\alpha.$$
> The torque of this force about the axis is
> $$	au_i = F_i\, r_i = m_i r_i^{2}\,\alpha.$$
> Summing over all particles of the body: …

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.What do you mean by torque? Derive an expression for torque in terms of moment of inertia of a body about a given axis of rotation.

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.What do you mean by torque? Derive an expression for torque in terms of moment of inertia of a body about a given axis of rotation.

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