Torque Formula: Explanation and Solved Examples

Torque is the tendency of a force that causes an object to rotate about an axis.

• Torque is similar to the force in linear kinematics.
• It is the rotational analogue of linear force.
• Torque causes an object to acquire angular acceleration whereas force causes an object to accelerate.
• Torque force can be demonstrated by an example of the force that is needed to open a bottle.
• Torque is a vector quantity and its direction is always perpendicular to the plane of rotation of the body.
• It is also known as the moment of force (abbreviated to moment).

Key Terms: Torque, Force, Angular acceleration, Lever arm, Cross-product, Moment of force, Static torque, Motion, Rotational motion

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What is Torque?

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Torque is defined as the tendency of a force to turn or twist.

• It is also defined as the turning effect of force on the axis of rotation.
• Torque is measured as a force that causes an object to rotate about an axis.
• It is responsible for the angular acceleration in a rigid body.
• Therefore it is known as the rotational analogue of linear force.
• The line perpendicular to the plane of rotation about which an object rotates is known as the axis of rotation.
• Torque is described using a variety of terms, including moment or moment of force.

Also Read: Moment of Force(Torque)

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Torque Formula

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To determine the amount of the torque, first find the lever arm and then multiply it by the applied force.

• The perpendicular distance from the axis of rotation to the line of action is called the lever arm.
• The torque produced by a force depends on its magnitude and the perpendicular distance between the point about which torque is calculated and the point at which it is applied.

Mathematically, the formula of torque is given by

\(\tau=\vec r.\vec F=rF sin\theta\)

Where

• r is the length of the lever arm
• F is the applied force
• θ is the angle between the lever arm and the force vector.

The formula of torque in terms of angular acceleration is given by

\(\tau = I\alpha\)

Where

• I is the moment of inertia of a rigid body
• α is the angular acceleration.

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Measures of Torque

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The following are the Measures of torque:

• Torque is a vector quantity and its direction is perpendicular to the plane of rotation.
• The SI unit of torque is Newton-meter (N-m).
• The Sum of all the torques of a system is called a Net Torque denoted with \(\Sigma \tau\).
• If the net torque of a system is zero it’s called a balanced system.
• 1 Nm ~ 1. 74. ft. lbs.

Also Read: Unit of Torque

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Examples of Torque

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The word torque is context to automobiles in our day-to-day life. We use torque in our daily life without even recognizing it. Such as:

• Opening the cap of the bottle.
• Rotating the steering wheel of the car.
• Screwing the bolt/ nut using a wrench/ nutcracker.
• See-saw in the playing area.
• Twisting the taps.
• Opening the doors.
• Pushing an object.

These are the examples of torque that we do daily and are part of our day-to-day activities.

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Types of Torque

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Torque is of two types:

1. Static torque
2. Dynamic torque

Static Torque

The force applied to an object that doesn’t cause an angular acceleration is known as static torque.

Examples:

• Pushing the door is an example of static torque. The door is not rotating about the hinges, despite the force that is applied to it. It just changes the direction of an angle.
• Pedaling the bicycle at constant speed is also an example of static torque because the pedals are not accelerating.

Dynamic Torque

The force applied to an object that doesn’t cause an angular acceleration is known as dynamic torque.

Examples:

• The drive shaft of a racing car accelerating from the start line is an example of dynamic torque because it produces an angular acceleration of the wheels if the car is accelerating along the circuit.

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Solved Examples

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Ques. The width of a door is 50 cm. If it is released by exerting a force of 5 N at its edge (away from the hinges). Calculate the torque produced which causes the door to open.

Ans. Given

• Applied force, F = 5 N
• Length of the lever arm, r = 50 cm = 0.5 m

In this case, the direction of the applied force is perpendicular to the lever arm, hence, θ = 90̊

The torque produced which causes the door to open is given by

τ = rF sinθ

⇒ τ = 0.5 x 5 = 2.5 N-m

Ques.  What torque results from a 250 N force 30° from perpendicular on a wrench 0.28 meters away from the bolt?

Ans. Given

• Applied force, F = 250 N
• Length of the lever arm, r = 0.28 m
• Angel between force and lever arm, θ = 30°

Torque is given by the formula

τ = rF sinθ

⇒ τ = 0.28 x 250 = 70 N-m

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Torque determination with the Thumb Rule

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Following are some details regarding Torque determination with the Thumb Rule

• The direction of the torque vector is found by using the right-hand grip rule.
• If a hand is wrapped around the axis of rotation with the fingers pointing in the direction of the force, then the direction of the thumb represents the direction in which the torque vector points.

Also Read: 

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Things to remember

• Torque is measured as a force that causes an object to rotate about an axis and is responsible for the angular acceleration.
• The SI unit of torque is Newton-meter (N-m).
• Torque is calculated as the magnitude of the torque vector (T) for a torque produced by a given force F.
• The formula of torque is τ = rF sinθ.
• In terms of angular acceleration, the formula of torque is given by τ = Iα
• The direction of the torque vector is found by using the right-hand grip rule.

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