Derviation of Moment of inertia

Click For Summary

Discussion Overview

The discussion revolves around the concept of moment of inertia (MI) in circular motion, exploring its definition, relationship with mass and radius, and the reasons for its mathematical representation as mr² rather than mr. Participants express confusion regarding the dependence of certain physical quantities on the square of another term and the implications of this in the context of rotational dynamics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why moment of inertia is represented as mr² instead of mr, expressing a lack of understanding about inertia and its implications in circular motion.
  • Another participant clarifies that moment of inertia depends on the radius from the axis of rotation but is not proportional to it; rather, it is proportional to the square of the radius.
  • A participant draws a parallel to Newton's second law, questioning why force is not expressed as ma², indicating confusion about the different dependencies of physical quantities.
  • One response distinguishes between Newton's laws and moment of inertia, explaining that moment of inertia is analogous to mass in linear motion, affecting how torque relates to angular acceleration.
  • Another participant expresses ongoing confusion about why some quantities depend on the square of a given quantity while others depend linearly on it.
  • A later reply introduces a mathematical perspective, suggesting that the term "moment" implies an additional factor related to the radius in the context of rotational dynamics.
  • One participant provides a derivation involving a particle in circular motion, linking linear and angular quantities to arrive at the formula for torque in terms of moment of inertia.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and remains unresolved regarding the conceptual understanding of moment of inertia and the reasons behind its mathematical formulation.

Contextual Notes

Participants express confusion over the definitions and relationships between different physical quantities, indicating a need for clarity on the underlying principles of rotational dynamics and the mathematical representations involved.

anmolnanda
Messages
20
Reaction score
0
our lecturer defined MI as inertia of body in circular motion(i don't know anything about inertia)
except that it is something to oppose
my question is that
when M.I in circular motion depends on both mass and radius(distance from axis) of the body.
why MI is represented as mr^2 why is it not mr
how can some quantity depend on the square of another term when they are also directly prop. to that quantity without square...
 
Physics news on Phys.org
Why do you think it should be mr? Momentum of inertia *depends* on the radius from the axis of rotation, but is not *proportional* to the radius. In fact it is proportional to the square of the radius. It is defined this way so that (momentum of inertia) times (angular velocity) = (angular momentum).
 
my question is why is the square
force=ma
not
force=ma^2
 
Newton's law is a completely separate equation. Perhaps you are confused by the terminology "moment of inertia"?

Newton's second law is (force) = (mass) * (acceleration). It tells you the rate of change of an object's velocity if you apply a given force to it.

Moment of inertia is defined by (moment of inertia) = (mass) * (distance from axis of rotation)^2. In conjunction with the equation (torque) = (moment of inertia) * (angular acceleration) it let's you find the rate of change of angular velocity if you apply a torque. Moment of inertia is kind of an analog to mass: if something has a large mass, you have to apply a large force to accelerate it much. If something has a large moment of inertia, you have to apply a large torque to get it to rotate very quickly.
 
i am just confused or i mean trying to figure out why some quantities depends on square of a given quantity and some degree of 1 of it...
 
anmolnanda said:
i am just confused or i mean trying to figure out why some quantities depends on square of a given quantity and some degree of 1 of it...

hi anmolnanda! :smile:

"moment of …" means "r cross …", so anything with moment has an extra r :wink:

(and you can take any formula like F = ∫ a dm, do "r cross" of the whole thing, and get another formula …

in this case, r x F = ∫ r x a dm = ∫ r x (r x α) dm, or τ = Iα)
 
Suppose a particle of mass m is attached to a pivot by a thin rod of length r . As the particle travels around the circle, we know that the distance it travels is equal to the angle the rod sweeps out measured in radians multiplied by the radius r . Differentiating twice shows that
a = r A

where A is the angular acceleration (i.e. the rate at which the angular velocity of the rod is changing) and a is the instantaneous linear acceleration the particle experiences out on the circle.

By Newton's second law for linear motion, if we apply a force F to the particle, then F = m a . On the other hand, since we have a rotating system, we would like to work with torque , instead of force, so we multiply both sides of the equation by r . Then

T = F r = m r a .

Finally, we use the equation derived about, to convert from linear acceleration to angular acceleration:

T = m r a = m r (A r ).

Rearranging terms gives the desired formula T = (m r 2) A.
 

Similar threads

Undergrad Moment of Inertia about an axis and Torque about a point
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Moment of inertia of human body with limbs at an angle
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Axial angular momentum calculation
  • · Replies 30 ·
2
Replies
30
Views
5K
Graduate How can I calculate the moment of inertia of any object?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad About torque and moment of inertia
  • · Replies 30 ·
2
Replies
30
Views
8K
Graduate Torque and Moment of Inertia of a Lever Arm
  • · Replies 2 ·
Replies
2
Views
4K
Graduate About gymnastic spins (moment of inertia related)
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Why Can't Moment of Inertia Exceed MR² for Solid Bodies?
  • · Replies 27 ·
Replies
27
Views
8K
Undergrad What is the Moment of Inertia? A 10 Minute Introduction
  • · Replies 0 ·
Replies
0
Views
4K
Undergrad Moment of inertia and angular KE confusion
  • · Replies 8 ·
Replies
8
Views
2K