IEMDaily - Video Lecture Notes

Definition

Rigid Body

A rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it.

Example: A metal rod in an example of rigid body.

Formula

Instantaneous angular velocity of rigid body

Instantaneous angular velocity:
The Instantaneous rate at which an object rotates in a circular path.

ω = lim_{Δ t \to 0} \frac{Δ θ}{Δ t}

ω = \frac{d θ}{d t}

Definition

Angular velocity of a rigid body from linear velocity of two points

Let two points on a rigid body have position vectors

{\vec{r}}_{1}

and

{\vec{r}}_{2}

and velocity vectors

{\vec{v}}_{1}

and

{\vec{v}}_{2}

. Then, angular velocity can be found using:

{\vec{v}}_{2} - {\vec{v}}_{1} = ({\vec{r}}_{2} - {\vec{r}}_{1}) \times \vec{ω}

Example

No-slip condition of string on pulley

Example:
A mass

m

is attached to a pulley through a cord as shown in the figure. The pulley is a solid disk with radius

R

. The cord does not slip on the disk. The mass is released from rest at a height

h

from the ground and at the instant the mass reaches the ground, the disk is rotating with angular velocity

ω

. The mass of the disk is

M = x m (\begin{matrix} \frac{2 g h}{R^{2} ω^{2}} - 1 \end{matrix})

. Find

x

.

Solution:
Let the tension be Tnow

m g - T = m a

..(1)

T R = \frac{M R^{2} α}{2}

....(2)

a = α R

...(3)
from above equations we get

a = \frac{m g}{\frac{M}{2} + m}

now final velocity

V^{2} = 2 a h = \frac{2 m g h}{\frac{M}{2} + m}

Also

V^{2} = ω^{2} R^{2}

M = 2 m (\frac{2 g h}{R^{2} ω^{2}} - 1)

Definition

Relation Between Angular Velocity and Linear Velocity

Relation Between Angular Velocity and Linear Velocity:

v = w r

Definition

Relation between frequency, time period and angular velocity

w = \frac{2 π}{T} = 2 π ν

rad/s

Definition

Radius of Gyration

Radius of gyration refers to the distribution of the components of an object around an axis. In terms of mass moment of inertia, it is the perpendicular distance from the axis of rotation to a point mass (of mass, m) that gives an equivalent inertia to the original object(s).

R_{g}^{2} = \frac{I}{m}

Definition

Real life examples of moment of inertia

FLYWHEEL of an automobile: Flywheel is a heavy mass mounted on the crankshaft of an engine. The magnitude of MOI of the flywheel is very high and helps in storing the energy.
Hollow shaft- An hollow shaft transmits more power compared to that of a solid shaft(both of same mass). The MOI of the hollow shaft is more compared to that of a solid shaft.
Shipbuilding- Moment of inertia has a big impact on shipbuilding. A ship may sink by rolling but a ship will never sink by pitching.

Definition

Define Moment of inertia

Definition:
Moment of inertia of a rigid body about an axis of rotation is defined as the sum of product of the mass of each particle and the square of its perpendicular distance from the axis of rotation.

∴ I = \sum_{i = 1}^{n} m_{i} r_{i}^{2}

Definition

Physical Significance of Moment of inertia

An unbalanced force produces linear motion, whereas unbalanced torque produces rotational motion. To produce linear motion in a body the unbalanced force is applied to overcome inertia. In this case inertia of a body is called as mass, which depends upon the amount of matter concentrated in the body.

F = M a

..........(1)

To produce rotational motion in a body an unbalanced torque is applied to overcome its inertia. In this case inertia of a body is called the rotational ineria on moment of inertia.

τ

I α

..........(2)

Comparing equations we can say that moment of inertia plays same role in rotational motion as mass dose in linear motion.

The mass of body always remains constant, but moment of inertia depends upon position of axis of rotation.