Rotational Motion of a Rigid Body in Physics

Rotational Motion Of Rigid Body

Edited By Vishal kumar | Updated on Jul 02, 2025 07:11 PM IST

Rotational motion is a fundamental aspect of physics that describes the movement of objects spinning around a fixed axis. When we observe a ceiling fan rotating, a spinning top, or even the Earth revolving around its axis, we are witnessing rotational motion in action. Unlike linear motion, where an object moves along a straight path, rotational motion involves objects moving in circles or around a central point. In the context of a rigid body, which does not deform under the forces applied to it, the entire body moves in a synchronized manner, maintaining its shape and size. This concept is crucial in understanding the behaviour of various machines, from simple wheels to complex gears in engines, and plays a significant role in disciplines like engineering and robotics. Understanding rotational motion not only helps us grasp the mechanics of everyday objects but also enables us to design and innovate in ways that make our lives easier and more efficient. In this article, we will discuss the rotational motion of the rigid body, its examples, and the definition of rotational Motion.

This Story also Contains

What is a Rigid Body?
What is Translation Motion?
What is Rotational Motion?
Some Important Terms Used in Rotational Motion
Solved Examples Based on Rotational Motion
Summary

Rotational Motion Of Rigid Body

What is a Rigid Body?

A rigid body is an idealized concept in physics that refers to an object with a fixed shape and size, where the distances between any two points within the object remain constant, regardless of the forces or torques applied to it. In other words, a rigid body does not deform, bend, or compress under the influence of external forces.

What is Translation Motion?

Translational motion refers to the movement of an object in which all its points move in the same direction and cover the same distance in a given period. This type of motion occurs when an object shifts from one position to another along a straight or curved path without rotating. If a body is moving such that a line drawn between any two of its internal points remains parallel to itself.

All the particles of the body move along parallel paths.
All the particles of the body follow 1 D motion.

NEET Highest Scoring Chapters & Topics

This ebook serves as a valuable study guide for NEET exams, specifically designed to assist students in light of recent changes and the removal of certain topics from the NEET exam.

Download E-book

Example- Motion of a body along a straight line.

What is Rotational Motion?

A rigid body is said to be in pure rotation if every particle of the body moves in a circle and the centre of all the circles lies on a straight line called the axis of rotation. The line joining any two internal points does not remain parallel.

Example-motion of wheels, gears, motors.

Some Important Terms Used in Rotational Motion

What Is the System?

A system is a collection of any number of particles interacting with one another and is under observation for analysing the situation.

What are Internal Forces?

Internal forces are all the forces exerted by various particles of the system on one another. Internal forces between two particles are equal in magnitude and opposite in direction.

What are External Forces?

External forces are the forces that we have to apply on the object/system from outside to move or stop the object/system.

Solved Examples Based on Rotational Motion

Example 1: The internal forces between two particles are

1) Equal in magnitude only

2) Equal in magnitude and direction both

3) Equal vectors

4) None of the above

Solution :

Internal forces

Internal forces are all the forces exerted by various particles of the system on one another. Internal forces between two particles are equal in magnitude and opposite in direction.

Hence, the answer is the option (1).

Example 2: In the case of pure rotation, if we join any two internal points by a line then -

1) They will intersect each other

2) It will remain parallel always

3) It will form a cycloidal path

4) None of the above

Solution:

When a body undergoes pure rotation then the line joining any two points also rotates which will cause the intersection of the line joining both points.

Hence, the answer is the option (1).

Example 3: In the rotational motion of a rigid body, all particles move with

1) same linear and angular velocity

2) same linear and different angular velocity

3) with different linear velocities and the same angular velocities

4) with different linear velocities and different angular velocities

Solution:

Rotation

A rigid body is said to be in pure rotation if every particle of the body moves in a circle and the centre of all the circles lies on a straight line called the axis of rotation.

wherein

The line joining any two internal points does not remain parallel.

In a rigid body

all particles rotate with the same angular velocity but different linear velocities

$\vec{v} = \vec{w} \times \vec{r}$

r is different for all particles

Hence, the answer is the option (3).

Example 4: A metal coin of mass 5g and radius of 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure. The system is initially at rest. the constant torque, which will make the system rotate about AB at 25 rotations per second in 5s, is close to:

1) $4.0 \times 10^{- 6} Nm$
2) $1.6 \times 10^{- 5} Nm$
3) $7.9 \times 10^{- 6} Nm$
4) $2.0 \times 10^{- 5} Nm$

Solution:

$\begin{aligned} α = \frac{Δ ω}{Δ t} = \frac{25 \times 2 π}{5} = 10 π rad / s^{2} \\ τ = (\frac{5}{4} M R^{2}) α \\ = \frac{5}{4} \times 5 \times 10^{- 3} \times {(10^{- 2})}^{2} \times 10 π \\ ≃ 2 \times 10^{- 5} Nm \end{aligned}$

Hence, the answer is the option (4).

Summary

This article covers the essential concepts of rotational motion in rigid bodies, distinguishing it from translational motion. It explains the key terms like the system, internal forces, and external forces, providing clarity on the principles that govern rotational dynamics. The article also includes solved examples that illustrate these concepts in action, making the understanding of rotational motion more practical and applicable.

Frequently Asked Questions (FAQs)

1. How does the concept of torque relate to rotational motion?

Torque is the rotational equivalent of force in linear motion. It causes a rigid body to rotate or change its rotational motion. Torque is calculated as the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force.

2. What is the moment of inertia, and how does it affect rotational motion?

The moment of inertia is a measure of an object's resistance to rotational acceleration. It depends on the object's mass distribution relative to the axis of rotation. Objects with greater moment of inertia require more torque to achieve the same angular acceleration.

3. What is angular velocity, and how is it different from linear velocity?

Angular velocity is the rate of change of angular position with respect to time, measured in radians per second. Unlike linear velocity, which describes motion along a straight line, angular velocity describes rotational motion around an axis.

4. How does the parallel axis theorem help in calculating moment of inertia?

The parallel axis theorem states that the moment of inertia of an object about any axis parallel to an axis passing through its center of mass is equal to the moment of inertia about the center of mass axis plus the product of the object's mass and the square of the perpendicular distance between the two axes.

5. How does the radius of rotation affect the relationship between angular and linear velocity?

The linear velocity of a point on a rotating rigid body is equal to the product of its angular velocity and the radius of rotation. As the radius increases, the linear velocity increases proportionally, while the angular velocity remains constant for all points on the rigid body.

6. What is a rigid body in rotational motion?

A rigid body in rotational motion is an object that maintains a fixed shape and size as it rotates around an axis. All points in the body move in circular paths around this axis, with the distance between any two points remaining constant throughout the rotation.

7. How does the distribution of mass in a rigid body affect its rotational motion?

The distribution of mass in a rigid body affects its moment of inertia. Objects with mass concentrated farther from the axis of rotation have a larger moment of inertia, making them more resistant to changes in rotational motion. This is why, for example, a hollow cylinder is harder to start or stop rotating than a solid cylinder of the same mass and radius.

8. What is the rotational kinetic energy of a rigid body?

The rotational kinetic energy of a rigid body is the energy associated with its rotational motion. It is calculated as 1/2 * I * ω², where I is the moment of inertia and ω is the angular velocity. This is analogous to the linear kinetic energy formula, with moment of inertia replacing mass and angular velocity replacing linear velocity.

9. What is precession, and how does it relate to rotational motion?

Precession is the slow rotation of the axis of a spinning object around another axis due to an external torque. It occurs when a torque is applied perpendicular to the spin axis of a rotating object, causing the axis to rotate around the torque axis. This phenomenon explains the motion of gyroscopes and the precession of the Earth's rotational axis.

10. How does the concept of rolling without slipping apply to rigid bodies?

Rolling without slipping occurs when a rigid body rolls along a surface with no relative motion between the point of contact and the surface. In this case, the linear velocity of the center of mass is equal to the product of the angular velocity and the radius of the object. This concept is important in understanding the motion of wheels and other rolling objects.

11. What is the difference between rolling and sliding motion for a rigid body?

Rolling motion occurs when a rigid body rotates about an axis while its center of mass moves in a straight line. In pure rolling, there is no slipping between the object and the surface it rolls on. Sliding motion, on the other hand, involves the object moving along a surface without rotating or with rotation that doesn't match the translation. The key difference is that rolling involves both rotational and translational motion, while sliding is purely translational.

12. How does the concept of torque apply to couples?

A couple is a pair of equal and opposite forces acting on a rigid body, separated by a perpendicular distance. The torque produced by a couple is equal to the product of one of the forces and the perpendicular distance between them. Couples are important because they produce pure rotation without translation, regardless of the point about which torque is calculated.

13. What is the difference between rotational kinetic energy and translational kinetic energy in rolling motion?

In rolling motion, an object possesses both rotational and translational kinetic energy. The translational kinetic energy is associated with the motion of the center of mass and is calculated as 1/2 * m * v², where m is the mass and v is the linear velocity of the center of mass. The rotational kinetic energy is associated with the rotation about the center of mass and is calculated as 1/2 * I * ω², where I is the moment of inertia and ω is the angular velocity. The total kinetic energy of a rolling object is the sum of these two components.

14. How does the concept of rotational equilibrium apply to rigid bodies?

Rotational equilibrium occurs when the net torque acting on a rigid body is zero. This means that the body is either not rotating or rotating at a constant angular velocity. For a body to be in complete equilibrium, it must be in both translational and rotational equilibrium. Understanding rotational equilibrium is crucial for analyzing static structures and balancing rotating machinery.

15. How does the rotational motion of a rigid body affect its center of mass?

The center of mass of a rigid body undergoing pure rotational motion follows a circular path around the axis of rotation. The radius of this path is the perpendicular distance from the axis of rotation to the center of mass. Understanding this concept is crucial for analyzing the motion of rotating objects like wheels, gears, and celestial bodies.

16. How does the conservation of angular momentum apply to rotating rigid bodies?

The conservation of angular momentum states that in the absence of external torques, the total angular momentum of a system remains constant. For a rigid body, this means that as its moment of inertia changes, its angular velocity must change in the opposite direction to maintain constant angular momentum.

17. What is the relationship between torque and angular momentum?

Torque is the rate of change of angular momentum with respect to time. In other words, the net torque applied to a rigid body is equal to the time derivative of its angular momentum. This relationship is analogous to the relationship between force and linear momentum in linear motion.

18. What is the moment of inertia tensor, and why is it important for 3D rotations?

The moment of inertia tensor is a 3x3 matrix that describes how an object's mass is distributed in three-dimensional space. It is important for 3D rotations because it allows us to calculate the moment of inertia about any arbitrary axis of rotation. This is crucial for analyzing complex rotational motions in three dimensions, such as those encountered in spacecraft dynamics or robotics.

19. How does the concept of angular impulse relate to changes in angular momentum?

Angular impulse is the rotational analog of linear impulse. It is defined as the product of torque and the time interval over which it acts. The angular impulse applied to a rigid body is equal to the change in its angular momentum. This concept is useful for analyzing situations where torques act for short durations, such as in collisions or impacts involving rotating objects.

20. How does the parallel axis theorem simplify calculations of moment of inertia for complex shapes?

The parallel axis theorem allows us to calculate the moment of inertia of an object about any axis parallel to an axis passing through its center of mass. This simplifies calculations for complex shapes by allowing us to break them down into simpler components. We can calculate the moment of inertia for each component about its own center of mass, then use the parallel axis theorem to find the moment of inertia about the desired axis of rotation.

21. How does the principle of angular momentum conservation apply to planetary motion?

The conservation of angular momentum is fundamental to understanding planetary motion. In the absence of external torques, the angular momentum of a planet orbiting the sun remains constant. This principle explains why planets move faster when they are closer to the sun (perihelion) and slower when they are farther away (aphelion). As the radius of orbit decreases, the planet's angular velocity must increase to conserve angular momentum, resulting in elliptical orbits as described by Kepler's laws.

22. What is the relationship between angular momentum and spin in quantum mechanics?

In quantum mechanics, angular momentum and spin are closely related but distinct concepts. Angular momentum in quantum systems is quantized, meaning it can only take on certain discrete values. Spin is an intrinsic form of angular momentum possessed by elementary particles, which doesn't arise from rotational motion in the classical sense. The total angular momentum of a quantum system includes both orbital angular momentum (similar

23. What is angular acceleration, and how is it related to torque?

Angular acceleration is the rate of change of angular velocity with respect to time. It is directly proportional to the net torque applied to the object and inversely proportional to the object's moment of inertia, as described by the rotational form of Newton's Second Law: τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration.

24. How does the principle of parallel axis theorem extend to three dimensions?

In three dimensions, the parallel axis theorem states that the moment of inertia tensor about any axis can be calculated by adding the moment of inertia tensor about a parallel axis passing through the center of mass and the product of the object's mass and the square of the perpendicular distance between the two axes. This allows for easier calculation of moments of inertia for complex shapes and orientations.

25. What is the rotational work-energy theorem, and how does it differ from its linear counterpart?

The rotational work-energy theorem states that the work done by torques on a rigid body is equal to the change in its rotational kinetic energy. It is expressed as W = ΔKE_rot, where W is the work done by torques and ΔKE_rot is the change in rotational kinetic energy. This is analogous to the linear work-energy theorem, but it uses torque and angular displacement instead of force and linear displacement.

26. What is the relationship between angular momentum and moment of inertia for a rigid body?

The angular momentum of a rigid body is the product of its moment of inertia and angular velocity: L = Iω, where L is angular momentum, I is moment of inertia, and ω is angular velocity. This relationship shows that for a given angular velocity, objects with larger moments of inertia have greater angular momentum.

27. What is the significance of the radius of gyration in rotational motion?

The radius of gyration is a characteristic length that represents the distribution of mass in a rotating object. It is defined as the distance from the axis of rotation to a point where, if all the mass of the object were concentrated, the moment of inertia would be the same as that of the actual object. The radius of gyration is useful for comparing the rotational inertia of different objects and simplifying calculations involving moment of inertia.

28. How does the concept of angular momentum conservation apply to figure skaters?

Figure skaters utilize the conservation of angular momentum to increase their rotation speed. When a skater pulls their arms and legs close to their body during a spin, they decrease their moment of inertia. Since angular momentum (L = Iω) must remain constant in the absence of external torques, the decrease in moment of inertia (I) results in an increase in angular velocity (ω), making the skater spin faster.

29. What is the role of friction in the rotational motion of rigid bodies?

Friction plays a crucial role in the rotational motion of rigid bodies, particularly in rolling motion. Static friction between a rolling object and the surface prevents slipping, allowing for pure rolling motion. Without friction, objects would slide rather than roll. Additionally, friction can provide the torque necessary to start or stop rotation, such as when applying brakes to a wheel.

30. How does the concept of torque explain the operation of a wrench or lever?

A wrench or lever operates based on the principle of torque. The torque applied to a bolt or object is equal to the force applied multiplied by the perpendicular distance from the axis of rotation to the point where the force is applied. By increasing the length of the wrench or lever arm, we can increase the torque applied for a given force, making it easier to turn a bolt or lift a heavy object.

31. What is the relationship between angular displacement, angular velocity, and angular acceleration?

The relationships between angular displacement (θ), angular velocity (ω), and angular acceleration (α) are analogous to those in linear motion. Angular velocity is the rate of change of angular displacement with respect to time: ω = dθ/dt. Angular acceleration is the rate of change of angular velocity with respect to time: α = dω/dt. For constant angular acceleration, we can use equations similar to those in linear kinematics, replacing linear quantities with their angular counterparts.

32. How does the rotational motion of the Earth affect the motion of objects on its surface?

The Earth's rotational motion affects objects on its surface in several ways. It causes the apparent deflection of moving objects (the Coriolis effect), which influences weather patterns and ocean currents. It also creates a centrifugal force that slightly counteracts gravity, making objects slightly lighter at the equator than at the poles. Additionally, the Earth's rotation causes the alternation of day and night and influences the tides in combination with the Moon's gravitational pull.

33. What is the principle behind gyroscopic stability, and how is it applied in real-world situations?

Gyroscopic stability is based on the conservation of angular momentum. A rapidly spinning object resists changes to its axis of rotation due to its large angular momentum. This principle is applied in various real-world situations, such as in bicycle wheels to maintain balance, in ship stabilizers to reduce rolling motion, and in spacecraft attitude control systems to maintain orientation in space.

34. How does the concept of rotational inertia apply to the design of flywheels in engines?

Rotational inertia, or moment of inertia, is crucial in the design of flywheels. Flywheels are used in engines to store rotational energy and smooth out fluctuations in angular velocity. A flywheel with a large moment of inertia (achieved by concentrating mass at the rim) can store more energy for a given angular velocity. This helps maintain a steady rotation of the engine's crankshaft, even when the power strokes of the pistons are not continuous.

35. What is the relationship between torque and power in rotational motion?

Power in rotational motion is the rate at which work is done or energy is transferred. It is equal to the product of torque and angular velocity: P = τω, where P is power, τ is torque, and ω is angular velocity. This relationship is analogous to the power equation in linear motion (P = Fv) and is important in understanding the performance of rotating machinery, such as motors and turbines.

36. How does the concept of precession explain the motion of a spinning top?

Precession in a spinning top occurs due to the torque exerted by gravity. As the top spins rapidly about its axis, the gravitational torque causes the axis to rotate slowly around the vertical, tracing out a cone. The rate of precession is inversely proportional to the spin rate of the top. This is why a rapidly spinning top appears to stand upright, with its axis precessing slowly, while a slowly spinning top quickly falls over.

37. What is the role of moment of inertia in the design of rotating machinery?

Moment of inertia plays a crucial role in the design of rotating machinery. It affects the machine's resistance to changes in rotational speed, energy storage capacity, and vibrational characteristics. Engineers must carefully consider the moment of inertia when designing components like flywheels, turbine rotors, and motor armatures to optimize performance, efficiency, and stability. A larger moment of inertia can provide more stable rotation but requires more torque to change speed.

38. What is the significance of the moment of inertia tensor in analyzing complex rotational motions?

The moment of inertia tensor is crucial for analyzing complex rotational motions, especially in three dimensions. It provides a complete description of an object's resistance to rotational acceleration about any axis. The tensor allows us to calculate the moment of inertia about any arbitrary axis of rotation, which is essential for understanding the dynamics of objects that rotate about multiple axes or have changing axes of rotation, such as spacecraft, gyroscopes, or complex mechanical systems.

39. How does the concept of rotational motion apply to the functioning of centrifuges?

Centrifuges utilize rotational motion to separate substances of different densities. As the centrifuge rotates at high speed, the centripetal acceleration causes denser substances to move outward from the axis of rotation more than less dense substances. The separation force in a centrifuge is proportional to the square of the angular velocity and the radius of rotation. Understanding these principles of rotational motion is crucial for designing and operating centrifuges effectively in various applications, from medical laboratories to industrial processes.