Moment Of Inertia And It’s Physical Significance, Important Factors, Parallel, Perpendicular Axis Theorem
Moment of inertia : According to Newton’s first law of motion, inertia of a body is defined as the inability of a body to change by itself its state of rest or of linear motion. Similarly, in rotational motion any body continues its state of motion unless applied by an external torque to change that state.
Shrivastav 30 Nov, 2023
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This property of a body by virtue of which it opposes its state of rest or of uniform rotation about an axis is called rotational inertia.
Definition :
The moment of inertia of a rigid body is defined as the sum of the product of masses of the particles constituting a body and the square of their respective distances from the axis of rotation.
Physical significance of moment of inertia
Greater the moment of inertia of the body, greater the torque required to change its state of motion. The moment of inertia plays same role in rotational motion as mass plays in linear motion. That is why it is called rotational analogue of mass in linear motion.Factors on which the moment of inertia depends :
The moment of inertia is a measure of the manner in which it’s different parts are distributed at different distances from the axis of rotation. Unlike mass, it is not a fixed quantity but rather depends on position and orientation of the axis of rotation with respect to the body as a whole. (i) Mass of the body (ii) size and shape of the body (iii) Distribution of mass about the axis of rotation. (iv) Position and orientation of the axis of rotation with respect to the body.Check : Important Chapters and Topics to Rank Higher In JEE
Daily life examples based on moment of inertia
(i) Despite being turned off, the electric fan continues to run for some time. This is because the blades of the fan were moving. After being switched off, they will take time to rest. (ii)The swirling motion continues after you stop stirring coffee or tea. (iii)In bicycle, bullock cart ,etc., the moment of inertia is increased by concentrating most of the mass at the rim of the wheel and connecting the rim to the axle through spokes.Even after we stop paddling , wheels of the bicycle continue to rotate for sometime due to large moment of inertia.Also Check : JEE Main Marks vs Percentile
Parallel Axis Theorem In moment Of Inertia
The parallel axis theorem refers to the moment of inertia of an object in an axis parallel to the body, and passing through its centre is the overall moment of inertia of a rigid object. It is about the axial direction stationed at the centre and the product of the body’s mass times the square of a difference between two axes. The formula is as follows, I = I cm + Mh 2 In the formula, I depict the MOI of the body, and Ic denotes the MOI of the centre. M stands for mass and h 2 for the square of the distance among both axes.Also Check : JEE Main Eligibility Criteria 2024
Perpendicular axis theorem In moment Of Inertia
It states, the moment of inertia or rotational inertia of a planar item about an axis at a right angle to the surface is just the l amount of the rotational inertia of two axes passing through the same spot in the plane of the item. The formula is as follows I Z = I x + I y I denote the MOI, and the subscripts are the denotations of axes, respectively.Moment of inertia : FAQs
Q.1. What is the moment of inertia of a thin spherical shell?
Ans.The moment of inertia of a thin spherical shell is, I=2/3MR^2.
Q.2. What is the moment of inertia of a rod about an axis passing through the centre and perpendicular to its central axis? Given that mass of rod is 1kg, length = 10cm. a) 0.00083kgm2 b) 0.0833kgm2 c) 0.0033kgm2 d) 0.00033kgm2
Answer. - Let the moment of inertia be I.
And M & I be the mass and length of rod respectively.
For the given condition,
I = MI2/12
= 1*0.01/12 = 0.00083kgm2.
Q.3.Why we do prefer a wrench of a longer arm?
Ans- Torque is the product of force and perpendicular distance from its axis of rotation. Longer the handle, lesser is the torque required.
Q.4: Why is it easier to open a tap with two fingers than with one finger?
Ans-When we apply couple with single arm,an equal and opposite force of action acts at the axis of rotation and hence smaller couple is applied. So it is easier to open a tap with two fingers than single finger.
Q.5. On what factors does moment of inertia depends?
Ans- Moment of inertia depends on
(i) mass
(ii)perpendicular distance of body from the axis of rotation.
Q.6. Is moment of inertia scalar or vector quantity?
Ans- Scalar quantity.
Q.7 : Mention the physical quantity that corresponds to the moment of force.
Ans- The moment of force is called torque.
Q.8: What is the S.I. unit of moment of inertia?
Ans- S.I. unit is kgm2
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