Short Answer:
The relation between torque and angular acceleration states that the torque acting on a rotating body is directly proportional to the product of its moment of inertia and angular acceleration. This means that when a torque is applied to a body, it produces an angular acceleration depending on how much rotational resistance (moment of inertia) the body has.
Mathematically, the relation is expressed as:
where, T is torque, I is the moment of inertia, and α is the angular acceleration. This equation is similar to Newton’s second law of motion in rotational form.
Detailed Explanation :
Relation between Torque and Angular Acceleration
The relation between torque and angular acceleration explains how an applied torque causes a change in the rotational motion of a body. Just like in linear motion where force produces acceleration, in rotational motion, torque produces angular acceleration. The amount of angular acceleration produced depends on the moment of inertia of the rotating body, which represents its resistance to rotational motion.
This relation forms the foundation of rotational dynamics and is used to understand how different torques affect rotating machinery, engines, wheels, and other mechanical systems.
Definition of Torque
Torque (T) is the turning effect of a force acting at a distance from the axis of rotation. It causes an object to rotate about that axis. The magnitude of torque is given by:
where,
T = torque (N·m)
F = applied force (N)
r = perpendicular distance from the axis of rotation to the line of action of the force (m).
If multiple forces act on a body, the resultant torque determines the net effect on its rotation. Torque can be thought of as the rotational equivalent of force.
Definition of Angular Acceleration
Angular acceleration (α) is the rate of change of angular velocity with respect to time. It shows how quickly a rotating body speeds up or slows down. It is expressed as:
where,
α = angular acceleration (rad/s²)
ω = angular velocity (rad/s).
If a torque acts continuously on a body, it changes the angular velocity, and hence angular acceleration occurs.
Moment of Inertia and Its Role
The moment of inertia (I) of a body measures its resistance to angular acceleration. It depends on the mass distribution of the body about its axis of rotation. A higher moment of inertia means that more torque is required to achieve the same angular acceleration. It is similar to mass in linear motion.
The moment of inertia is given by:
where m is the mass of each particle and r is its perpendicular distance from the axis of rotation.
Derivation of the Relation
Consider a rigid body rotating about a fixed axis under the action of an external torque T.
Let a small element of the body of mass m be at a distance r from the axis of rotation.
The tangential force acting on this element is:
where a is the tangential acceleration of the element.
But, a is related to angular acceleration α as:
Substituting this value of a in the force equation,
Now, the torque acting on the particle is:
For the entire body, the total torque is the sum of the torques on all mass elements:
Since the term is the moment of inertia (I) of the body, we get:
This is the fundamental relation between torque and angular acceleration.
Physical Meaning of the Relation
The equation shows that:
- Torque (T) is directly proportional to angular acceleration (α).
- The constant of proportionality is the moment of inertia (I).
This means that for a given torque, if the moment of inertia is large, the angular acceleration will be small, and vice versa.
For example, a small wheel (low I) spins faster under the same torque compared to a large flywheel (high I).
Comparison with Linear Motion
The relation in rotational motion is similar to Newton’s second law in linear motion, .
| Linear Motion | Rotational Motion |
| Force (F) | Torque (T) |
| Mass (m) | Moment of Inertia (I) |
| Linear Acceleration (a) | Angular Acceleration (α) |
This similarity helps in understanding rotational motion as an angular form of linear motion.
Examples in Engineering Applications
- Electric Motors:
When torque is applied to the rotor of a motor, it produces angular acceleration that determines the speed of rotation. - Vehicle Wheels:
Torque from the engine creates angular acceleration in the wheels, causing the vehicle to move forward. - Flywheels and Rotating Shafts:
Engineers use this relation to calculate the angular acceleration produced by a given torque in flywheels or turbine shafts. - Machine Design:
Designers determine the required torque to rotate machine parts with known moment of inertia at a desired angular acceleration.
Practical Understanding
- If T increases and I remains constant, α increases proportionally — meaning the body spins faster.
- If I increases (heavier or larger body) while T remains constant, α decreases — the body spins slower.
- Zero torque results in no angular acceleration, meaning the body rotates at a constant speed.
These observations are important in controlling rotational systems and understanding their energy efficiency.
Conclusion
The relation between torque and angular acceleration is given by the equation . It forms the basis of rotational dynamics and shows that torque produces angular acceleration depending on the body’s moment of inertia. This principle is widely used in mechanical engineering applications like engines, turbines, and rotating machinery to design systems that can effectively control and utilize rotational motion.