Relation Between Torque And Power

The relation between torque and power plays an essential role in the design, operation, and optimization of mechanical systems.

• Torque, also known as moment of force, is the tendency of a force to cause the rotation of the body to which it is applied. 
• The torque is equal to the amount of the force vector component located in the plane perpendicular to the axis multiplied by the shortest distance between the axis and the direction of the force component concerning the axis of rotation.
• The SI unit of torque is Newton-meter (N m).
• Power is a measure of the rate at which work is done.
• The SI unit of power is watt (W).

The relation between the torque and the power is given by the formula

P = \(\tau\) x ω

Where

• P is the power i.e. work done per unit time.
• \(\tau\) is the magnitude of the torque
• ω is the angular velocity

Key Terms: Angular velocity, Torque, Power, Force, Linear displacement, Angular speed, Moment of force, Horsepower, Work done, Energy

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What is Torque?

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Torque is the force that may cause an object to rotate around an axis.

• It is the rotational equivalent of linear force.
• It is also known as the moment of force (abbreviated to moment). 
• It expresses the rate of change of angular momentum supplied to an isolated body.
• Torque is defined as the product of the magnitude of the perpendicular component of a force and the distance of the force's line of action from the location being determined. 
• Torque can be explained using the concept of conservation of energy.
• Torque is a vector quantity and its direction is always perpendicular to the plane containing the force and displacement vector.
• The SI unit of torque is Newton-meter (N m).
• The dimensional formula of torque is [M L² T^-2]

Torque Formula

The amount of torque delivered to a rigid body is determined by three variables: the applied force, the lever arm vector connecting the point of torque measurement to the point of force application, and the angle between the force and lever arm vectors.

It is given by

\(\vec{\tau} = \vec{r} \times \vec{F}\) or \(\tau =rF sin\theta\)

Where

• \(\tau\) is the magnitude of the torque
• r is the magnitude of the position vector
• θ is the angle between the force vector and lever arm vector.

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What is Power?

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Power is defined as the time rate at which work is done or energy is transferred.

• Power is also defined as the amount of force required to produce a unit displacement.
• Power is a scalar quantity.
• The SI unit of power is watt (W). One watt is equal to one joule per second.
• Horsepower (hp) is a unit of measurement for power or the rate at which work is done, generally about engine or motor output.
• 1 Horsepower is equivalent to 745.7 watts.
• The dimensional formula of power is [M L² T^-3].

Power

Power Formula

The formula of power is given by

P = \(\frac{E}{t}\) or P = \(\frac{W}{t}\)

Where

• P is the power
• E is the energy consumed to do work
• W is the work done
• t is the time during which work is done.

In terms of force and the velocity produced by the force, the formula of power can be given as

P = F x v

Where

• F is the force applied to an object
• v is the velocity of the object in the direction of applied force.

Other formulas of power are

• P = VI
• P = I²R
• P = V²/R

Where

• I is the current flowing through a conductor
• R is the resistance of the conductor
• V is the potential difference across the conductor.

Horsepower

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Relation between Torque and Power Formula

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The formula that shows the relationship between the torque and the power is given by

P = \(\tau\) x ω

Where

• P is the power i.e. work done per unit time.
• \(\tau\) is the magnitude of the torque
• ω is the angular velocity

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Derivation of Relation between Torque and Power

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To get the relationship between torque and power, the rotatory motion of a body is compared to its linear equivalent. The linear displacement of a rotating body is the distance covered at the circumference of the rotation. It is given by the product of the angle covered and radius.

The relation between the angular speed and linear speed of a rotating body is given by

v = rω

Where

• v is the linear speed of the body
• r is the radius of the circular path
• ω is the angular speed

But v = (linear distance d)/(time t)

⇒ Linear distance, d = rωt ….(i)

Torque of a rotating body is given by

\(\tau\) = F x r

⇒ Force, F = \(\tau\)/r …(ii)

Work done by the force to produce a linear distance of d is given by

W = F x d

Power delivered, P = W/t = Fd/t

Using equation (i), we get

P = (F x rωt)/t

Using equation (ii), we get

P = \(\tau\) x ω

Hence, the product of the magnitude of the torque and angular velocity given power.

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Solved Examples

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Ques. A constant torque of 25 N m is needed to make a wheel make 10 revolutions in 20 seconds. What is the average power exerted by this torque?

Ans. Given

• The magnitude of the torque, \(\tau\) = 25 N m
• Frequency of the rotation of the wheel, f = 10/20 = 0.5 Hz

Angular frequency of the rotation is given by

ω = 2πf

On substituting the values, we get

ω = 2 x 3.14 x 0.5 = 3.14 rad/s

Now the relation between the torque and power is given by

P = \(\tau\) x ω

Hence the average power exerted by the torque is given by

P = 25 x 3.14 = 78.5 W

Ques. An electric motor exerts a constant torque of 5 N m on a flywheel by which it is rotated at the rate of 420 rpm. What is the power of the motor?

Ans. Given

• The magnitude of the torque, \(\tau\) = 5 N m
• Frequency of the rotation, f = 420 rpm = 7 rps

The relationship between the torque and power is given by

P = \(\tau\) x ω

Where ω is the angular frequency.

But, ω = 2πf = 2 x 3.14 x 7 = 43.96 rad/s

On substituting the values of torque and angular frequency, we get

P = 5 x 43.96 = 220 W

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Things to Remember

• Torque is the rotational equivalent of linear force that may cause an object to rotate around an axis.
• Power is defined as the time rate at which work is done or energy is transferred.
• Power is a scalar quantity, whereas torque is a vector quantity.
• The direction of the torque is always perpendicular to the plane containing the force and displacement vector.
• The SI unit of power is watt (W).
• 1 Horsepower is equivalent to 745.7 watts.
• The relation between the angular speed (ω) and linear speed (v) of a rotating body is given by v = rω.
• The relation between the torque (\(\tau\)) and the power (P) is given by P = \(\tau\) x ω.

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