Question

Discuss the following topics:

1. Rotational Dynamics : Angular displacement, Angular velocity, angular acceleration.

2. Rigid Body and Moment of Inertia

3. Angular Momentum, conservation of Angular Momentum and Torque with some examples of applications.

4. Mechanical Equilibrium and conditions of equilibrium

5. Elasticity, Plasticity and Hooke's Law

6. Parallel Forces and net torque

7. Coefficient of Elasticity, stress, and strain.

Answer 1

Angular displacement is defined as “the angle in radians (degrees, revolutions)" through which a point or line has been rotated in a specified sense about a specified axis.

OR

Angular displacement is defined as the shortest angle between the initial and the final positions for a given object having a circular motion about a fixed point. It is a vector quantity. Thus it will have the magnitude as well as the direction. The direction is represented by a circular arrow pointing from the initial position to the final position. It may be either clockwise or anti-clockwise in direction

If s is the distance travelled by the body, and r is the radius of the circle along which it is moving.

then angular displacement θ=s/r,

In simpler words, the displacement of object is the distance travelled by it around the circumference of a circle divided by its radius.

Angular velocity:angular velocity ω is defined as the rate of change of an angle. i.e. ω=Δθ/Δt

where an angular rotation Δθ takes place in a time Δt. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ω is analogous to linear velocity v. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length Δs in a time Δt, and so it has a linear velocity v=Δs/Δt  

but Δθ=Δs/r

therefore Δs = rΔθ.

Substituting this into the expression for v gives, v = rΔθ/Δt = rω   

where ω = Δθ/Δt is the angular velocity

Angular acceleration

The rate of change of angular velocity regarding time is angular acceleration, which is a vector quantity. It is denoted by α. The angular acceleration formula is given by,

Definition: Angular acceleration of an object undergoing circular motion is defined as the rate with which its angular velocity changes with time. Angular acceleration is also referred to as rotational acceleration. It is a vector quantity, that is, it has both magnitude and direction. Angular acceleration is denoted by α and is expressed in the units of rad/s2 or radians per second square. Formula: Angular acceleration can be expressed as given below, α=dω/dt And also in terms of the double differentiation of the angular displacement, as given below, α=d2θ/dt2 Derivation: Angular acceleration is the rate of change of angular velocity with respect to time, or we can write it as, α=dω/dt Here, α is the angular acceleration that is to be calculated, in terms of rad/s2, ω is the angular velocity given in terms of rad/s and t is the time taken expressed in terms of seconds. Angular velocity as we know can be expressed as given below. ω=vr Here, ω is the angular velocity in terms of rad/s, v is the linear velocity and r is the radius of the path taken. Angular Velocity can also be expressed as the change in angular displacement with respect to time, as given below. ω=dθ/dt Where θ is the angular rotation of the object and t is the total time taken. Using the above formula, we can write angular acceleration α as α=d2θ/dt2

Angular displacement