Question

Establish the relation between moment of inertia and torque.
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Answer 1

Answer:

When a torque is applied to an object it begins to rotate with an acceleration inversely proportional to its moment of inertia. This relation can be thought of as Newton's Second Law for rotation. The moment of inertia is the rotational mass and the torque is rotational force. Angular motion obeys Newton's First Law.

Explanation:

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Answer 2

⍟ Relation btw Torque &MI :-

Let an object be in rotatory motion with mass m, moving along an arc of a circle with radius r,under action of torque.

We know:

$\leadsto \sf a=\frac{f}{m}$ (from newton's 2nd law)

❥When a torque acts on a body which is in rotatory motion,it produces angular acceleration.Hence,

Substitute linear acceleration a with angular acceleration.

We know :

$\leadsto \sf a=\frac{d}{dt} (\frac{ds}{dt} ).....(1)$

As the body is in rotatory motion,

=> s=rdθ

Substituting value of 's' in (1):

$\\ :\implies \sf a=\frac{d}{dt} (\frac{rd\theta }{dt} )\\\\:\implies \sf a=r\frac{d}{dt} (\frac{d\theta }{dt} )\\\\:\implies \sf \bold{ a=r\alpha }$

❥ Torque is tendency of force to turn.

Hence,replacing force by torque we get :

$\implies \sf \tau =Fr\\\\\implies \sf F=\frac{\tau }{r}$

Now,

Substituiting the value of 'F'&'a' in newtons second law we get,

$:\implies \sf r\alpha =\frac{F}{m} \\\\:\implies \sf r\alpha =\frac{(\tau /r)}{m}\\\\:\implies \bold{\tau =mr^2 \alpha }$

We know:

❥ Moment of inertia (I)=mr²α

Hence,the above equation becomes,

$\mapsto \boxed{\boxed{\sf \tau =I\alpha }}$

Where,  

• τ is Torque.
• I is the moment of inertia.
• α is angular acceleration .

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✪ Know More:-

✦ Torque :Torque is a measure of how much a force acting on an object causes that object to rotate.

✦ Moment of inertia :It is a rotating body's resistance to angular acceleration or deceleration, equal to the product of the mass and the square of its perpendicular distance from the axis of rotation.

✦ Unit of torque id Newton-meter.

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