Work, Energy And Power For Rotating Body

Edited By Vishal kumar | Updated on Jul 02, 2025 06:14 PM IST

Understanding how things move and work is important in everyday life. When we talk about rotating bodies, we mean objects that spin around, like the wheels on a bike, the blades of a fan, or even the Earth itself. Just like pushing a cart requires effort (or work), making something spin also requires work. This work gets stored as energy in the rotating object. The faster or harder it spins, the more energy it has. Power is about how quickly this energy is used or transferred.

Work, Energy And Power For Rotating Body

Think about a windmill: the wind works to turn the blades, storing energy in their rotation, which can then be converted into electricity to power our homes. Or consider a car: the engine does work to make the wheels spin, giving the car energy to move. By exploring these concepts of work, energy, and power in rotating bodies, we can better understand how many of the tools and machines we use every day operate and make our lives easier.

What is Work?

For translation motion W=∫Fds

So for rotational motion W=∫τdθ

What is Rotational kinetic energy?

The energy a body has by virtue of its rotational motion is called its rotational kinetic energy.

Difference between Rotational kinetic energy and Translatory kinetic energy

Rotational kinetic energy	Translatory kinetic energy
KR=12Iω2	KT=12mV2
KR=12Lω	KT=12PV
KR=L22I	KT=P22m

What is Power?

Power is equal to the rate of change of kinetic energy. It's denoted by P.

For translation motion P=F→⋅V→

So for rotational motion

P=d(KR)dt=d(12Iω2)dt=Iωdωdt=Iαω=τ⋅ω Or P=τ→⋅ω→

We can understand better through video.

Solved Example Based on Work, Energy and Power for Rotating Body

Example 1: A stationary horizontal disc is free to rotate about its axis. When a torque is applied to it, its kinetic energy as a function of θ, where θ is the angle by which it has rotated, is given as kθ2. If its moment of inertia is I then the angular acceleration of the disc is :

1) k4Iθ
2) kIθ
3) k2Iθ
4) 2kIθ

Solution:

kE=kθ212Iω2=kθ2ω2=2kθ2Ia=dωdθ2ωdωdθ=2k2θIdωdθ=2kθIωa=2kθIω

angular acceleration
α=aω=(2kθIω)=2kθI

Hence, the answer is the option (4).

Example 2: A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is :
1) L/4
2) 2L
3) 4L
4) L/2

Solution:

We know that L=Iω
K.Erot =12Iω2LK=2IωIω2⇒>LK=2ω=>L=2Kω∴L1L2=K1K2×ω2ω1=2×2=4L1L2=4⇒>L2=L14=L4

Hence, the answer is the option (1).

Example 3: A wheel is rotating freely with an angular speed ω on a shaft. The moment of inertia of the wheel is I and the moment of inertia of the shaft is negligible. Another wheel of the moment of inertia 3I initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is :
1) 56
2) 14
3) 0
4) 34

Solution:

By angular momentum conservation
ωI+3I×0=4Iω′⇒ω′=ω4(KE)i=12Iω2(KE)f=12×(4I)×(ω4)2=Iω28ΔKE=38 Lω2
fractional loss =ΔKEKE1=38Iω212 Tω2=34

Hence, the answer is option (4).

Example 4: A disc with a flat small bottom beaker placed on it at a distance R from its centre is revolving about an axis passing through the centre and perpendicular to its plane with an angular velocity ω. The coefficient of static friction between the bottom of the beaker and the surface of the disc is μ. The beaker will revolve with the disc if:

1) R≤μg2ω2
2) R≤μgω2
3) R≥μg2ω2
4) R≥μgω2

Solution:

The beaker will revolve with the disc if friction force is able to provide the necessary centripetal force

f≤μmgmv2R=mRω2⩽μmgR⩽μgω2

Hence, the answer is option (2).

Example 5: A disc of mass 1 kg and radius R is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of the disc. A body of the same mass as that of the disc is fixed at the highest point of the disc. Now the system is released when the body comes to the lowest position, its angular speed will be 4x3R rad s s−1 where x= (g=10 ms−2)
1) 5
2) 6
3) 7
4) 8

Solution:

As the system is released and comes to the lowest position

TEinitial =TEfinal Mgh+0=12I2Mg(2R)=12(MR22+MR2)ω28MgR=3MR2ω28 g3R=ω16x=8 g=80x=5

The value of X is 5

Summary

Work is done on a body only if the following two conditions are satisfied:

A force acts on the body
The point of application of the force moves in the direction of the force.

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There are two types of energy in a rigid body: kinetic energy and potential energy. The energy of a body is measured by the amount of work the body can perform. Energy has several forms.

Frequently Asked Questions (FAQs)

1. What is the rotational analog of power in linear motion?

The rotational analog of power is the rate at which work is done on a rotating body. It can be expressed as the product of torque and angular velocity: P = τω, where P is power, τ is torque, and ω is angular velocity. This is similar to linear power, which is the product of force and linear velocity.

2. How does rotational inertia affect the power required to accelerate a rotating body?

Rotational inertia, or moment of inertia, directly affects the power required to accelerate a rotating body. A larger moment of inertia means more power is needed to achieve the same angular acceleration. This is why it's harder to spin up a heavy flywheel compared to a lighter one with the same radius.

3. How does the concept of mechanical advantage apply to rotating systems?

Mechanical advantage in rotating systems relates to the ratio of output torque to input torque. It can be achieved through gears, pulleys, or levers. For example, a large gear turning a smaller gear will increase angular velocity but decrease torque, while a small gear turning a larger gear will increase torque but decrease angular velocity.

4. What is the rotational equivalent of mass in linear motion?

The rotational equivalent of mass in linear motion is the moment of inertia. Just as mass represents an object's resistance to linear acceleration, moment of inertia represents an object's resistance to angular acceleration. The larger the moment of inertia, the more torque is required to achieve a given angular acceleration.

5. How does the conservation of energy principle apply to rotating bodies?

The conservation of energy principle applies to rotating bodies just as it does to linear motion. The total energy (sum of kinetic, potential, and any other forms of energy) remains constant in a closed system. For a rotating body, this might involve conversions between rotational kinetic energy, gravitational potential energy, and other forms of energy.

6. What is rotational kinetic energy and how does it differ from linear kinetic energy?

Rotational kinetic energy is the energy associated with the rotation of an object about an axis. It differs from linear kinetic energy in that it depends on the object's moment of inertia and angular velocity, rather than mass and linear velocity. The formula for rotational kinetic energy is (1/2)Iω², where I is the moment of inertia and ω is the angular velocity.

7. What is the relationship between torque and rotational kinetic energy?

Torque is related to rotational kinetic energy through work. When a torque is applied to a rotating body, it performs work, which increases the body's rotational kinetic energy. The change in rotational kinetic energy is equal to the work done by the torque.

8. How does the moment of inertia affect a body's rotational energy?

The moment of inertia plays a crucial role in determining a body's rotational energy. A larger moment of inertia means that more energy is required to change the body's rotational motion. This is why objects with mass distributed farther from the axis of rotation have higher rotational kinetic energy for the same angular velocity.

9. How does the distribution of mass affect the rotational kinetic energy of an object?

The distribution of mass affects rotational kinetic energy through the moment of inertia. Objects with mass concentrated farther from the axis of rotation have a larger moment of inertia and thus higher rotational kinetic energy for the same angular velocity. This is why a hollow cylinder has more rotational kinetic energy than a solid cylinder of the same mass and radius when rotating at the same angular speed.

10. What is the work-energy theorem for rotational motion?

The work-energy theorem for rotational motion states that the work done on a rotating body is equal to the change in its rotational kinetic energy. Mathematically, it can be expressed as W = ΔKE_rot = (1/2)I(ω_f² - ω_i²), where W is work, KE_rot is rotational kinetic energy, I is moment of inertia, and ω_f and ω_i are final and initial angular velocities, respectively.

11. How does the concept of work apply to rotating bodies?

Work done on a rotating body is the product of the torque applied and the angular displacement. This is analogous to linear work, where force is replaced by torque and displacement by angular displacement. The formula for rotational work is W = τθ, where τ is the torque and θ is the angular displacement.

12. How does the concept of rotational work apply to a satellite in orbit?

For a satellite in a circular orbit, no rotational work is done by gravity because the gravitational force is always perpendicular to the direction of motion. However, if the orbit is elliptical, the varying gravitational force does perform work, changing the satellite's kinetic and potential energy as it moves through its orbit. This is described by Kepler's laws of planetary motion.

13. How does the concept of rotational work apply to a pendulum's motion?

In a pendulum's motion, rotational work is done by the torque due to gravity. As the pendulum swings down, the gravitational torque does positive work, increasing the pendulum's rotational kinetic energy. As it swings up, the torque does negative work, decreasing the kinetic energy and increasing potential energy. In an ideal pendulum, the total mechanical energy (sum of kinetic and potential) remains constant.

14. How does the concept of torque relate to rotational work and energy?

Torque is crucial in understanding rotational work and energy. It is the rotational analog of force and is responsible for changing the rotational motion of an object. Work done by a torque increases the rotational kinetic energy of the object. The relationship is given by the work-energy theorem for rotation: the work done by a torque equals the change in rotational kinetic energy.

15. What is the role of friction in the energy transformations of a rolling object?

Friction plays a crucial role in the energy transformations of a rolling object. Without friction, an object would slide rather than roll. Friction at the point of contact allows for the conversion of some translational kinetic energy into rotational kinetic energy. In the case of rolling without slipping, friction does no work and the total mechanical energy is conserved.

16. What is the relationship between linear and angular quantities in rotational motion?

Linear and angular quantities in rotational motion are related through the radius of rotation. For instance, linear velocity v = rω, where r is the radius and ω is the angular velocity. Similarly, linear acceleration a = rα, where α is the angular acceleration. These relationships allow us to convert between linear and rotational motion quantities.

17. What is the role of moment of inertia in determining the stability of rotating objects?

The moment of inertia plays a crucial role in determining the stability of rotating objects. Objects with a larger moment of inertia about their axis of rotation are more stable and resistant to changes in their rotational motion. This is why a spinning top with its mass distributed farther from the axis of rotation (larger moment of inertia) is more stable than one with its mass concentrated near the axis.

18. What is the significance of the moment of inertia in the context of angular momentum conservation?

The moment of inertia is crucial in understanding angular momentum conservation. When angular momentum is conserved (L = Iω = constant), changes in moment of inertia (I) lead to changes in angular velocity (ω). This principle explains phenomena like a figure skater spinning faster when pulling their arms in (decreasing I) or a diver somersaulting faster when tucking (decreasing I).

19. What is the significance of the parallel axis theorem in calculating rotational energy?

The parallel axis theorem is crucial for calculating the rotational energy of objects rotating about an axis that doesn't pass through their center of mass. It states that the moment of inertia about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the perpendicular distance between the axes.

20. How does angular momentum relate to rotational kinetic energy?

Angular momentum (L) and rotational kinetic energy (KE_rot) are related, but not directly proportional. Both depend on angular velocity (ω) and moment of inertia (I), but in different ways. Angular momentum is L = Iω, while rotational kinetic energy is KE_rot = (1/2)Iω². This means that doubling the angular velocity will double the angular momentum but quadruple the rotational kinetic energy.

21. What factors determine the rotational kinetic energy of a rolling object?

The rotational kinetic energy of a rolling object depends on its moment of inertia, its angular velocity, and its mass. The total kinetic energy of a rolling object is the sum of its translational kinetic energy ((1/2)mv²) and its rotational kinetic energy ((1/2)Iω²). The proportion of energy in each form depends on the object's shape and mass distribution.

22. What is the significance of the moment of inertia tensor in complex rotating systems?

The moment of inertia tensor is important for describing the rotational behavior of complex three-dimensional objects. It's a 3x3 matrix that relates the angular momentum of an object to its angular velocity. The tensor is crucial for understanding how objects rotate about arbitrary axes and how their rotation can change direction, which is especially important in fields like spacecraft dynamics and robotics.

23. How does the rotational kinetic energy of a system change during precession?

During precession, the rotational kinetic energy of a system remains constant, but its distribution changes. The total angular momentum is conserved, but the axis of rotation changes orientation. This leads to a complex motion where the energy is redistributed among different axes of rotation, while the total rotational kinetic energy remains the same.

24. How does the rotational kinetic energy of a planet affect its shape?

The rotational kinetic energy of a planet significantly affects its shape. Faster rotating planets tend to bulge at the equator and flatten at the poles due to the centrifugal effect. This shape distortion, known as oblateness, is a result of the balance between gravitational forces and the "centrifugal force" due to rotation. The amount of flattening depends on the planet's rotational speed, size, and mass distribution.

25. What is the relationship between angular momentum and rotational kinetic energy in a changing system?

While both angular momentum and rotational kinetic energy depend on angular velocity and moment of inertia, they change differently as a system evolves. Angular momentum is conserved in the absence of external torques, while rotational kinetic energy can change. For example, as a figure skater pulls in their arms, their moment of inertia decreases, angular velocity increases (conserving angular momentum), but rotational kinetic energy increases.

26. What is the significance of the parallel axis theorem in calculating the rotational energy of complex objects?

The parallel axis theorem is crucial for calculating the rotational energy of complex objects, especially those rotating about an axis that doesn't pass through their center of mass. It allows us to find the moment of inertia about any axis parallel to an axis through the center of mass. This is particularly useful for compound objects or when dealing with off-center rotations, as it simplifies the calculation of rotational kinetic energy.

27. How does the principle of conservation of angular momentum relate to changes in rotational kinetic energy?

The conservation of angular momentum can lead to changes in rotational kinetic energy. When angular momentum is conserved (L = Iω = constant), a change in moment of inertia (I) must be accompanied by a change in angular velocity (ω). Since rotational kinetic energy is (1/2)Iω², this change can result in an increase or decrease in rotational kinetic energy, even though angular momentum remains constant.

28. What is the relationship between torque and angular acceleration in rotational dynamics?

The relationship between torque and angular acceleration is analogous to force and linear acceleration in linear dynamics. It's given by the equation τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration. This equation, known as the rotational form of Newton's Second Law, shows that the angular acceleration produced by a torque is inversely proportional to the moment of inertia.

29. How does the rotational kinetic energy of a rigid body change during precession?

During precession, the total rotational kinetic energy of a rigid body remains constant, but its distribution among different axes of rotation changes. The precession itself doesn't change the magnitude of the angular velocity vector, but it does change its direction. This results in a complex motion where energy is transferred between different components of rotation while conserving the total rotational kinetic energy.

30. How does the distribution of mass in a rotating object affect its angular acceleration under a given torque?

The distribution of mass in a rotating object affects its angular acceleration through the moment of inertia. For a given torque, objects with mass concentrated closer to the axis of rotation (smaller moment of inertia) will experience greater angular acceleration. This is why a solid disk accelerates more slowly than a ring of the same mass and radius when subjected to the same torque.

31. What is the relationship between work done by friction and the changes in rotational and translational kinetic energy of a rolling object?

For a rolling object, the work done by friction at the point of contact is generally zero if there's no slipping. However, friction is responsible for converting some of the object's translational kinetic energy into rotational kinetic energy. The total mechanical energy remains constant, but energy is redistributed between translational and rotational forms to maintain the rolling motion.

32. How does the concept of rotational inertia apply to the design of flywheels for energy storage?

Rotational inertia is crucial in flywheel design for energy storage. Flywheels with larger moments of inertia can store more rotational kinetic energy for a given angular velocity. This is why flywheels designed for energy storage often have their mass concentrated at the rim, maximizing the moment of inertia. The energy stored can then be extracted by slowing the flywheel down when needed.

33. What is the significance of the parallel axis theorem in calculating the rotational energy of a compound pendulum?

The parallel axis theorem is essential for calculating the rotational energy of a compound pendulum. It allows us to determine the moment of inertia about the pivot point, which is not at the center of mass. By using this theorem, we can calculate the total rotational kinetic energy of the pendulum, which is crucial for understanding its period of oscillation and energy exchanges during its motion.

34. How does the principle of conservation of energy apply to a rotating body in a gravitational field?

For a rotating body in a gravitational field, the principle of conservation of energy states that the sum of rotational kinetic energy, translational kinetic energy, and gravitational potential energy remains constant in the absence of non-conservative forces. This principle governs the motion of objects like satellites, where energy is continually exchanged between these forms as the object moves through its orbit.

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 the rate at which energy is transferred. It's given by 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 shows that power can be increased by either increasing torque or angular velocity.

36. How does the rotational kinetic energy of a rigid body change during nutation?

During nutation, which is a small, periodic variation in the axis of rotation of a spinning object, the total rotational kinetic energy of the rigid body remains approximately constant. However, there is a small exchange of energy between the spinning motion and the nutation itself. This energy exchange causes the nutation amplitude to vary slightly, but the overall effect on total rotational kinetic energy is typically negligible.

37. What is the significance of the moment of inertia tensor in describing the rotational energy of asymmetric objects?

The moment of inertia tensor is crucial for describing the rotational energy of asymmetric objects because it accounts for the distribution of mass in all three dimensions. For such objects, the rotational kinetic energy depends on the orientation of the rotation axis relative to the principal axes of the object. The tensor allows us to calculate the rotational energy for any arbitrary axis of rotation, which is essential for understanding complex rotational motions.

38. How does the concept of work apply to a torque that varies with angular displacement?

When a torque varies with angular displacement, the work done is calculated by integrating the torque over the angular displacement. Mathematically, this is expressed as W = ∫τ(θ)dθ, where W is work, τ(θ) is the torque as a function of angular displacement θ. This is analogous to the work done by a varying force in linear motion and is crucial for understanding energy transfers in systems with non-constant torques.

39. What is the role of rotational kinetic energy in the physics of gyroscopes?

Rotational kinetic energy plays a crucial role in gyroscope physics. The high rot