Short Answer:

Torque is the turning effect of a force applied on a body about an axis of rotation. It is the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force.

The relation between torque and angular acceleration is given by Newton’s Second Law for rotational motion, which states that the torque acting on a body is directly proportional to the angular acceleration produced in it and the moment of inertia of the body. Mathematically, it is expressed as

where  = torque,  = moment of inertia, and  = angular acceleration.

Detailed Explanation :

Torque and its Relation with Angular Acceleration

In mechanics, when a force acts on a body and tends to rotate it about a fixed axis, the rotational effect of that force is called torque. It is an important concept in the study of rotational motion, just as force is important in linear motion. Torque depends on the magnitude of the force, the distance from the axis, and the angle at which the force is applied.

The relation between torque and angular acceleration is derived from Newton’s Second Law of Motion for rotational systems. It establishes that the torque acting on a rotating body is equal to the product of its moment of inertia and the angular acceleration produced.

Definition of Torque

Torque is defined as the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force.

Mathematically,

where,
= torque (N·m),
= applied force (N),
= perpendicular distance from axis (m).

Torque is a vector quantity and acts along the axis of rotation. Its direction is determined by the right-hand rule — if the fingers of the right hand curl in the direction of rotation caused by the force, then the thumb points in the direction of torque.

Physical Meaning of Torque

Torque represents the rotational ability of a force. A larger torque means a greater ability of the force to cause rotation. For example, using a longer spanner to open a tight nut is easier because it increases the perpendicular distance , thereby increasing the torque for the same applied force.

Thus, torque can be considered as the rotational equivalent of force in linear motion.

Derivation of the Relation Between Torque and Angular Acceleration

Consider a rigid body rotating about a fixed axis under the action of a force .
Let,

•  = mass of a small particle of the body,
•  = perpendicular distance of the particle from the axis of rotation,
•  = angular acceleration of the body,
•  = tangential acceleration of the particle.

Now, tangential acceleration is related to angular acceleration as

From Newton’s Second Law, the tangential force acting on the particle is:

The torque on this particle about the axis is:

For the whole rigid body consisting of many particles, the total torque is the sum of torques on all particles:

Since angular acceleration  is the same for all particles of the rigid body,

Here,  is defined as the moment of inertia (I) of the body about the axis of rotation.

Hence,

This is the fundamental relation between torque and angular acceleration.

Interpretation of the Relation

• The equation  is the rotational form of Newton’s Second Law of Motion.
• It shows that torque is directly proportional to the angular acceleration of the body.
• The constant of proportionality is the moment of inertia (I), which represents the rotational resistance of the body.

Hence, for a given torque:

• A body with large moment of inertia will experience small angular acceleration.
• A body with small moment of inertia will experience large angular acceleration.

This explains why it is easier to spin a bicycle wheel than a heavy flywheel using the same torque.

Units of Torque

• SI Unit: Newton-meter (N·m)
• CGS Unit: Dyne-centimeter (dyne·cm)
• Dimensional Formula:

Examples of Torque in Daily Life

1. Opening a Door:
   When you push a door at its handle, the applied force produces torque that rotates the door about its hinges. Pushing closer to the hinges requires more force because the distance (r) is smaller.
2. Turning a Bolt with a Wrench:
   A longer wrench provides a larger torque because it increases the perpendicular distance from the axis.
3. Rotating a Wheel:
   The engine of a vehicle provides torque to rotate the wheels, producing angular acceleration and motion.
4. Fan or Turbine Blades:
   Torque produced by motor or fluid pressure causes angular acceleration in the blades.
5. Crankshaft in Engines:
   Torque generated by combustion in the cylinder causes rotation of the crankshaft.

Practical Applications in Mechanical Engineering

1. Design of Machines:
   Torque analysis helps determine the power and efficiency of rotating machinery like turbines, pumps, and engines.
2. Vehicle Performance:
   The torque generated by an engine determines the vehicle’s ability to accelerate and carry loads.
3. Structural Analysis:
   In shafts and beams, torque helps engineers analyze stresses and design for safety.
4. Control of Rotating Systems:
   Used in gyroscopic systems, flywheels, and robotic arms to control angular acceleration and stability.

Conclusion

Torque is the rotational effect of a force that tends to rotate a body about a fixed axis. It depends on the magnitude of the applied force and the perpendicular distance from the axis of rotation. The relation between torque and angular acceleration is given by , where  is the moment of inertia. This equation shows that torque is directly proportional to angular acceleration, forming the foundation of rotational dynamics. The concept is widely used in mechanical systems involving rotation such as engines, turbines, and machines.