# Dynamics Of Rotational Motion About A Fixed Axis

We have read about the dynamics of particle motion before where we discovered that they were capable of undergoing translational motion only, but as we know, rigid bodies can undergo translational as well as rotational motion. So, in such cases, both the linear and the angular velocity need to be analysed.

In order to simplify these problems, we define the translational and rotational motion of the body separately. The rotational motion of the object is referred to as the rotational motion of an object about a fixed axis. In the next section, we will discuss the dynamics of rotational motion of an object about a fixed axis.

## Rigid Body Dynamics of Rotational Motion

A rotating body, as can be seen in the figure above, will have a point that has zero velocity, about which the object undergoes rotational motion. This point can be on the body or at any point away from it. Since the axis of rotation is fixed, we consider only those components of the torques applicable on the object that is along this axis as only these components cause rotation in the body. The perpendicular component of the torque will tend to turn the axis of rotation for the object from its position.

These results in the emergence of some necessary forces of constraint which finally tends to cancel the effect of these perpendicular components, thus restricting the movement of the axis from its fixed position, rendering its position to be maintained. Since the perpendicular components cause no effect; these components are not considered during the calculation. For any rigid body undergoing a rotational motion about a fixed axis, we need to consider only the forces that lie in planes perpendicular to the axis.

Forces which are parallel to the axis will give torques perpendicular to the axis and need not be taken into account. Also, only the components of position vector that are perpendicular to the axis are considered. Components of position vectors along the axis result in torques perpendicular to the axis and thus are not to be taken into account.

### How to calculate net force?

A uniform thin cylindrical disk of mass M and radius R is attached to two identical massless springs of spring constant k which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the springs and the axle are in the horizontal plane.

The unstretched length of each spring is L.

The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall.

The disk rolls without slipping with velocity Vo.

Coefficient of friction is µ.

Suppose we need to find the force when the disc is displaced by a distance ‘x’.

It is said that the disc moves without slipping. So, the frictional force is static and hence we cannot say f = μ N, instead we consider it to be ‘f’.

Also, a = α R —– (1)

From free body diagram, 2kx – f = ma —— (2)

f.R = Iα  —— (3)

Where, I = mR22 Using (1), (2) and (3)

we get, Net force = 4kx3

## Rotational Motion and Work-Energy Principle

According to the work-energy principle, the total work done by the sum of all the forces acting on an object is equal to the change in the kinetic energy of the object.

In rotational motion, the concept of the work-energy principle is based on the torque. It is stated as the object is said to be in a balanced state if its displacements and rotations are equal to zero work when a force is applied.

Consider a rigid body such that Δ𝛳 is the small rotation experienced by the object. Then the linear displacement is given as Δr = rΔ𝛳. This is perpendicular to the r.

Therefore, the work done is

ΔW = F perpendicular to Δr

ΔW = F Δr sin 𝜙

ΔW = Fr Δ𝛳 sin 𝜙

ΔW = 𝜏Δ𝛳

When the number of forces acting is increased, then the work done is given as

ΔW = (𝜏1 + 𝜏2 + ……) Δ𝛳

But we know that Δ𝛳 is the same for all forces.

Therefore, the work done will be zero, that is

𝜏1 + 𝜏2 + …… = 0

Hence, the work-energy principle for rotational motion is proved.

## Types of motion involving rotation

• Rotation about a fixed point
• Rotation which includes translational and rotational motion about an axis of rotation
• Rotation about an axis in the rotation

### Rotation about a fixed point

Ceiling fan rotation, rotation of the minute hand and the hour hand in the clock, and the opening and closing of the door are some of the examples of rotation about a fixed point.

### Rotation about an axis of rotation

Rotation about an axis of rotation includes translational as well as rotational motion. The best example of rotation about an axis of rotation is pushing a ball from an inclined plane. The ball reaches the bottom of the inclined plane through translational motion while the motion of the ball is happening as it is rotating about its axis which is rotational motion.

Another example of rotation about an axis of rotation is the motion of the earth. The earth rotates about its axis every day and it also rotates around the sun once every year. This is a classic example of translational motion as well as rotational motion.

## Rotational Dynamics

The rotational dynamics can be understood if you have ever pushed a merry-go-round. We observe that the change in the angular velocity of a merry-go-round is possible when a force is applied to it. Another example is the spinning of the bike wheel. As the force is increased, the angular acceleration produced in the wheel would be greater. Therefore, we can say that there is a relationship between the force, mass, angular velocity, and angular acceleration.

Consider a wheel of the bike. Let F be the force acting on the wheel such as angular acceleration produced is 𝛼. Let r be the radius of the wheel. We know that the force is acting perpendicular to the radius. We also know that,

F = ma

Where a is acceleration = r𝛼

Therefore,

F = mr𝛼

We have learned that torque is the turning effect of the force. Therefore,

𝜏 = Fr

rF = mr2𝛼

𝜏 = mr2𝛼

Therefore, we can say that the last equation is the rotational analog of F = ma such that torque is analog of force, angular acceleration is analog of acceleration, and rotational inertia that is mr2 is analog of mass. Rotational inertia is also known as moment of inertia.

The relationship between torque, the moment of inertia, and angular acceleration is

net 𝜏 = I𝛼

𝛼 = net 𝜏/I

Where net 𝜏 is the total torque

### Moment of inertia for symmetric bodies

Following is the table for a moment of inertia for symmetric bodies:

 Symmetric body Moment of inertia Ring with a symmetric axis I = mR2 Cylinder or disc with a symmetric axis $I=\frac{1}{2}mR^{2}$ Uniform sphere $I=\frac{2}{5}mR^{2}$ Rod with the axis through the end $I=\frac{1}{3}ml^{2}$ Rod with the axis at the center $I=\frac{1}{12}ml^{2}$

## Frequently Asked Questions – FAQs

### Are torque and moment of inertia similar?

No, torque and moment of inertia are not similar. Torque is dependent on the magnitude and direction of the force and on the application point. Whereas the moment of inertia is dependent on the mass and the axis of rotation.

### Define tangential acceleration.

Tangential acceleration, at is defined as the linear acceleration of a rotating object such that the linear acceleration is perpendicular to the radial acceleration. The SI unit of tangential acceleration is m/s2.

### What is the difference between angular acceleration and tangential acceleration?

Angular acceleration and tangential acceleration are most of the time considered to be similar, but they are not. Angular acceleration is defined as the change in the angular velocity of an object over time whereas tangential acceleration is defined as the change in the linear velocity of an object over time.

### Does angular acceleration change with radius?

No, angular acceleration does not change with radius. The angular acceleration remains constant throughout because the object moves as a rigid body with the same angle for the same amount of time.

### What is the difference between translational and rotational motion?

• The velocity of an object is constant when the object is moving under translational motion whereas the angular velocity of an object varies when the object is moving under rotational motion.
• In translational motion mass of an object is considered whereas in rotational motion moment of inertia of an object is considered.