What is the relation between torque and power?

Short Answer:

The relation between torque and power expresses how rotational force (torque) and rotational speed together determine the mechanical power transmitted by a rotating shaft or machine. Power is directly proportional to both torque and the angular speed of rotation.

In simple words, torque is the twisting force that causes rotation, while power is the rate at which work is done during rotation. The mathematical relation between them is given as:

where  is the power in watts,  is the torque in newton-meters, and  is the angular velocity in radians per second.

Detailed Explanation :

Relation Between Torque and Power

In mechanical systems, torque and power are closely related physical quantities that describe the performance of rotating machines like motors, engines, turbines, and gear systems. Torque represents the rotational force applied on a shaft, and power represents how quickly that torque can do work over time.

Whenever a shaft rotates under the influence of torque, mechanical power is produced or transmitted. The relationship between these two quantities helps engineers calculate the efficiency and output of machines used in industries, automobiles, and power plants.

Definition of Torque

Torque is the turning or twisting force that causes an object to rotate about an axis. It is the product of a force and the perpendicular distance from the axis of rotation to the line of action of the force.

Mathematically,

where,

  •  = Torque (N·m)
  •  = Force applied (N)
  •  = Radius or perpendicular distance (m)

Torque determines how much “rotational effort” is available to perform work. A higher torque means greater ability to overcome resistance or to start rotation.

Definition of Power

Power is defined as the rate of doing work. In rotational systems, it is the rate at which torque produces angular motion.

If a torque  acts on a rotating shaft and makes it rotate with angular velocity , then the power transmitted by the shaft is given by:

Here,

  •  = Power transmitted (Watts)
  •  = Torque (N·m)
  •  = Angular velocity (rad/s)

This relation shows that power depends on both how much torque is applied and how fast the rotation occurs.

Derivation of the Relation Between Torque and Power

Let a rotating shaft be driven by a torque  and rotating with an angular velocity .

  • The work done in one revolution of the shaft =  joules (since the shaft turns through  radians per revolution).
  • The number of revolutions per second (n) = , where  is the speed in revolutions per minute (rpm).

Now,

 

Therefore,

Where,

  •  = Power in watts (W)
  •  = Torque in newton-meters (N·m)
  •  = Speed in revolutions per minute (rpm)

This equation gives the mechanical power transmitted by a rotating shaft.

Alternate Forms of the Equation

  1. In kilowatts (kW):

or

  1. In horsepower (HP):

or approximately,

These forms are commonly used in motor and engine calculations where torque and speed are known.

Physical Meaning of the Relation

The relation  indicates that:

  • Power depends on both torque and speed.
  • For a constant torque, increasing the rotational speed increases power.
  • For a constant speed, increasing torque increases power.
  • If either torque or speed becomes zero, no power is transmitted.

For example, in a motor:

  • High torque at low speed means heavy lifting or pulling power (like in trucks).
  • Low torque at high speed means fast motion with low resistance (like in fans).

Thus, torque and speed balance determines how mechanical energy is utilized.

Example Calculation

Example:
A motor shaft delivers a torque of  at a speed of . Find the power transmitted.

Solution:

 

Thus, the shaft transmits 18.85 kW of power.

Practical Applications

  1. Electric Motors and Engines:
    Used to determine the power output for specific torque and speed values.
  2. Automobile Design:
    Helps in designing engines and gear systems for efficient performance.
  3. Machine Tools:
    Determines the torque required for cutting, drilling, or milling operations.
  4. Turbines and Generators:
    Used to measure mechanical power before converting it to electrical power.
  5. Industrial Shafts and Couplings:
    Ensures shafts are strong enough to safely transmit the required torque and power.

Importance of Torque–Power Relationship

  1. Design Efficiency:
    Engineers can optimize torque and speed combinations to achieve the desired power output.
  2. Performance Analysis:
    Determines how effectively a machine converts input energy into useful work.
  3. Load Matching:
    Helps in selecting motors and engines that provide sufficient torque for given power demands.
  4. Energy Conservation:
    Understanding this relation helps in reducing energy losses in mechanical transmission systems.
Conclusion

The relation between torque and power is fundamental in rotational mechanics. It establishes that power is the product of torque and angular velocity (). This means torque represents the turning force, while power measures how quickly work is done during rotation. This relation is crucial in designing, analyzing, and controlling mechanical systems like engines, motors, turbines, and shafts. By balancing torque and speed, engineers can ensure efficient and reliable power transmission in all types of mechanical equipment.