Derviation of Moment of inertia

In summary, the moment of inertia (in terms of mass and radius) of a rotating body is proportional to the square of the radius. This is important because it lets you find the rate of change of angular velocity if you apply a torque.
  • #1
anmolnanda
20
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...
 
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  • #2
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).
 
  • #3
my question is why is the square
force=ma
not
force=ma^2
 
  • #4
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.
 
  • #5
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...
 
  • #6
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)
 
  • #7
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.
 

FAQ: Derviation of Moment of inertia

What is the definition of moment of inertia?

The moment of inertia is a physical property of a rigid body that determines how difficult it is to change the body's rotational motion about a given axis. It is often referred to as the rotational mass.

How is the moment of inertia calculated?

The moment of inertia of a rigid body is calculated by summing the products of the mass and the square of the distance from each particle in the body to the axis of rotation.

What are the factors that affect the moment of inertia?

The moment of inertia is affected by the mass and distribution of mass of the rigid body, as well as the location and direction of the axis of rotation.

What is the equation for the moment of inertia of a point mass?

The moment of inertia of a point mass is equal to the mass multiplied by the square of the distance from the axis of rotation to the point mass (I = mr²).

How is the moment of inertia related to rotational kinetic energy?

The moment of inertia is a key factor in determining the rotational kinetic energy of a rigid body. The greater the moment of inertia, the more energy is required to change the rotational motion of the body.

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