Short Answer:
The work-energy principle for rotating bodies states that the work done by the torque acting on a rotating body is equal to the change in its rotational kinetic energy. In simple words, when a torque causes a body to rotate, it performs work that either increases or decreases the kinetic energy of rotation.
This principle connects the concepts of torque, angular displacement, and rotational kinetic energy. It helps in analyzing the motion of rotating systems such as wheels, shafts, and turbines. Mathematically, it is expressed as Work = Change in Rotational Kinetic Energy, i.e.,
Detailed Explanation :
Work-Energy Principle for Rotating Bodies
The work-energy principle for rotating bodies explains the relationship between the torque applied to a body and the resulting change in its rotational energy. Just as in linear motion, where work done by a force changes the kinetic energy of a body, in rotational motion, the work done by a torque changes the rotational kinetic energy of the body.
When a rigid body rotates about a fixed axis under the action of an external torque, the torque performs work on the body. This work results in a change in the body’s angular velocity, and hence its rotational kinetic energy.
Concept of Work in Rotational Motion
When a torque (T) acts on a rotating body and causes an angular displacement (θ), work is done.
The work done (W) in rotational motion is given by:
If the torque is not constant, then the small amount of work done is:
This equation shows that the torque in rotational motion plays the same role as force in linear motion, and the angular displacement plays the same role as linear displacement.
Rotational Kinetic Energy
When a body rotates, every particle in it moves in a circular path around the axis of rotation. The kinetic energy of each particle is due to its linear velocity. The total rotational kinetic energy (K) of the body is given by:
where,
I = moment of inertia of the body about the axis of rotation,
ω = angular velocity of the body.
This is the rotational equivalent of linear kinetic energy, which is given by .
Derivation of Work-Energy Principle
Let a rigid body rotate about a fixed axis under the action of a torque T.
When the body rotates through a small angle dθ, the small amount of work done is:
From the equation of rotational motion,
where α = angular acceleration.
Substituting this in the work equation,
Since,
and
Substituting,
Integrating between the limits of angular velocity ω₁ and ω₂:
Hence,
This proves the work-energy principle for rotating bodies.
Physical Meaning
The principle means that whenever torque does positive work on a body, the body’s rotational speed increases, and its kinetic energy increases. On the other hand, if the torque acts opposite to the direction of rotation, the work done is negative, and the rotational kinetic energy decreases.
For example, when a car accelerates, the torque applied by the engine does positive work and increases the rotational kinetic energy of the wheels. When brakes are applied, the torque due to friction does negative work, reducing the wheel’s rotational kinetic energy.
Applications in Mechanical Engineering
- Flywheels and Energy Storage:
Flywheels store rotational energy. The work done by the torque applied on a flywheel increases its rotational kinetic energy. - Rotating Shafts and Turbines:
The principle is used to calculate the energy transferred by torque in shafts, turbines, and motors. - Braking Systems:
The reduction of rotational energy in wheels during braking follows this principle. - Energy Analysis of Machines:
It helps engineers determine how much mechanical work is converted into rotational energy in machinery. - Design of Rotational Components:
It assists in calculating torque requirements for rotating parts to achieve desired speed and power.
Relation Between Linear and Rotational Motion
There is a close relationship between linear and rotational motion:
| Linear Motion | Rotational Motion |
| Force (F) | Torque (T) |
| Displacement (s) | Angular Displacement (θ) |
| Mass (m) | Moment of Inertia (I) |
| Linear Velocity (v) | Angular Velocity (ω) |
| Work (F × s) | Work (T × θ) |
| Kinetic Energy (½ m v²) | Rotational Kinetic Energy (½ I ω²) |
This comparison shows that rotational motion is the angular form of linear motion, and the work-energy principle holds true in both cases.
Importance of Work-Energy Principle
The work-energy principle for rotating bodies is very useful for solving problems related to rotational dynamics. It helps in calculating the torque required for a certain change in speed or determining the final speed when torque and angular displacement are known. It provides a simple and powerful method for analyzing energy changes in rotating systems without directly using equations of motion.
Conclusion
The work-energy principle for rotating bodies states that the work done by torque on a rotating body equals the change in its rotational kinetic energy. It is a vital concept in rotational dynamics, helping to understand how torque and angular motion are related to energy transfer. This principle is widely used in mechanical engineering applications like turbines, flywheels, and braking systems to analyze and design rotating machinery effectively.