Rotation Around 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.
FAQs on Rotation About a Moving Axis
1. What is meant by rotation about a moving axis?
Rotation about a moving axis describes the motion of a rigid body that is simultaneously rotating about an axis passing through its center of mass and translating (moving linearly) through space. This combined motion is also known as general plane motion. A classic example is a ball rolling on the ground.
2. How does rotation about a moving axis differ from pure rotational motion?
The key difference lies in the motion of the axis itself.
- In pure rotational motion, the body spins around a stationary or fixed axis, like a spinning top that stays in one spot. The axis of rotation does not move.
- In rotation about a moving axis, the body spins around an axis that is itself moving. For instance, the wheel of a moving bicycle rotates about its axle, and the axle moves forward with the bicycle.
3. What are some real-world examples of rotation about a moving axis?
This type of combined motion is very common in everyday life. Some key examples include:
- A bowling ball rolling down the lane.
- The wheels of a car or bicycle moving along a road.
- The planet Earth orbiting the Sun. Earth rotates on its own axis while the axis itself translates along its orbital path.
- A cylinder or drum rolling down an inclined plane.
4. Why is the center of mass so important for analysing this type of motion?
The concept of the center of mass is crucial because it allows us to simplify a complex problem. We can separate the general motion into two simpler, independent parts:
- The translational motion of the entire body, which is described as if all the mass were concentrated at the center of mass.
- The rotational motion of the body about an axis passing through its center of mass.
5. How do you calculate the total kinetic energy of a body rotating about a moving axis?
The total kinetic energy (K) is the sum of its translational and rotational kinetic energies. The formula, as per the CBSE syllabus for 2025-26, is:
K = Ktranslational + Krotational = ½MVc² + ½Iω²
Where:
- M is the total mass of the body.
- Vc is the linear velocity of the center of mass.
- I is the moment of inertia about the axis through the center of mass.
- ω (omega) is the angular velocity of the rotation.
6. What is the formula for the angular momentum of a body in this combined motion?
The total angular momentum (L) of a body rotating about a moving axis, with respect to a reference point, is the sum of the angular momentum of the center of mass and the angular momentum about the center of mass. The expression is:
L = Lc + (r × p)
Here:
- Lc is the angular momentum about the center of mass (also called spin angular momentum).
- r is the position vector from the reference point to the center of mass.
- p is the linear momentum of the body (p = MVc). The term (r × p) represents the orbital angular momentum.
7. In the case of a wheel rolling without slipping, what is the relationship between its linear and angular velocity?
For a wheel of radius R rolling without slipping on a flat surface, there is a direct relationship between the linear velocity of its center (Vc) and its angular velocity (ω). This condition is given by the formula:
Vc = Rω
This relationship is fundamental for solving problems involving rolling motion, as it connects the body's translational movement to its rotational movement. It signifies that the point of the wheel in contact with the ground is momentarily at rest.
