Rotation About a Moving Axis

There are different types of motions you have studied in the previous classes. One such motion is rotational motion. This is a remarkable type of motion where the direction of a body changes instantaneously and it rotates around an axis. The Rotation about a moving axis is something different. On this concept page, we will learn how a body can move in a rotational fashion around a moving axis. We will also calculate the force, speed, etc along with describing various examples.

What is Rotation?

Apart from the linear motion where the direction of a body does not change, we have studied motion where the direction changes but the speed remains the same. It is a type of motion where the body is moving around an axis at a uniform speed. It cannot be dictated as a velocity as it is changing direction at every instant. Hence, the calculation of force cannot be done when you cannot apply Newton’s Laws of Motion as the body is not moving on a linear path.

Rotation is different from that of revolution. Here, a body is not moving around an axis that does not pass through it. The rotation occurs when the axis passes through the body. The axis passes through the center of mass. If the center of mass is in a translation then the combination of these motions is called rotation about a moving axis.

What is Rotational Motion?

As we have discussed what rotation is, it becomes a lot easier to understand what rotational motion is. In this case, we will use a few rotational motion examples to make the discussion better.

Consider a rigid non-uniform body with a center of mass. If an axis passes through the center of mass and the body rotates on it, it is called rotational motion. When a top is spun and rotates stagnantly on the floor, it is called rotational motion. Here, you will find that the body is not translating on the floor. There is no linear motion. The only motion it has is the rotational motion. If you lift your bicycle and pedal, you will find only the back wheel is rotating but the cycle is not moving. It rotates around the axle where it is fixed. It is also an example of rotational motion.

Rotational Motion about a Moving Axis

Proceeding to the next level, when a rotating body is also translating, it is called rotation around a moving axis. If you have understood the concepts about rotation then consider the axis of this moving body is translating linearly. Let us consider the same example. When a top is spinning on its axis but the axis is also moving on a smooth floor. It is called rotational motion on a moving axis. This new concept gives birth to a whole new segment of physics called rotational mechanics.

How Can You Calculate Rotational Force and Torque?

In this case, we will have to understand and calculate two different momentums. The body of mass M will have a linear momentum for the translation and a rotational force. Hence,

Linear Momentum p = MVc 

(where Vc = linear velocity of the center of mass)

If there is any change in the momentum, a force has been applied as per Newton’s Second Law of Motion. Hence,

F = dp/dt

(where F = net force applied on the body)

Here comes the part where you have to calculate the angular momentum. Considering a reference point, the angular momentum of a body can be expressed as,

L = Lc + (r x p)

(where, Lc represents the angular momentum, r is the vector calculated from the reference point to the center of mass and p is the linear momentum)

You can see how the angular momentum is connected with the linear momentum as all the mass present in that body is attached to the center of mass. (r x p) is considered to be the orbital angular momentum. If there is any change in the angular momentum, it can be represented by the following equation.

𝜏 = dL/dt

The kinetic energy of this body M can be calculated as

K = ½(MVc2 + Iω2)

This is how rotational force is calculated. Follow this section to understand how the torque calculation for the rotating body is done too.

Example of Rotational Motion around a Moving Axis

You have now understood how to calculate and determine the rotational force of a rotating body. Let us consider a few examples to understand what this type of motion is and how it differs from rotational motion.

• When the earth rotates on its axis, it is called rotational motion. While doing this, its axis is also under translation. It is called rotational motion around a moving axis.

• When you are spinning a key ring in your finger and walking, the key ring’s axis is witnessing rotational motion around itself along with a translation due to your walking.

• When a drum rolls down an inclined plane. It witnesses rotating motion and its axis moves linearly.