Torque: Definition, Formula, Types, Problems and Examples

Torque in Physics: Torque is the measure of the twisting or rotational force that causes an object to rotate. We have all seen doors at our homes and schools. What makes the door rotate? It is clear that unless a force has applied, the door does not rotate. However, not every force is capable of doing this job. For example, a force applied to the hinge line of the door will not produce any rotation at all. In contrast, a force of a given magnitude applied at right angles to the door at its outer edge is most effective in producing rotation. It is because this force has a rotational effect on the hinge, known as torque.

Torque in Physics: Definition

Torque is the tendency of a force to cause or change the rotational motion of an object. Torque is also known as moment, or moment of force. Torque is a vector quantity, meaning it has both a direction and a magnitude. Torque is the measure of how much a force acting on an object causes it to rotate. It is the ‘twisting force’ that causes a body to rotate. The point or line about which a body rotates is called the axis of rotation. Thus, torque can be defined as the tendency of a force to turn or twist.

To rotate an object, it is not the force alone, but how and where the force is applied is important. The motion of a rigid body, in general, is a combination of rotation and translation. We know that a force is needed to change the translational state of an object, i.e., to produce linear acceleration. However, if the body is fixed at a point or along a line, it can only rotate, and in that case, the force must produce a rotational effect to rotate the body. Torque can be thought of as the analogue of force in the case of rotational motion.

Torque Formula

Torque refers to the twisting force that causes motion. The torque acting on a particle is defined as the product of the magnitude of the force acting on the particle and the perpendicular distance of the application of force from the axis of rotation of the particle. In vector form, torque is the cross product of the radius vector (from axis of rotation to the point of application of force) and the force vector. It is represented by the letter \(\vec \tau \). Its formula is given as:

\(\vec \tau = \vec r \times \vec F\)

Right-hand Rule: Determining the Direction of Torque

Torque is a vector quantity that is calculated using a cross or vector product. The torque is in the direction of the angular velocity produced by it, so the change in angular velocity is in the torque’s direction. To find the direction in which the torque acts on a given body, take your right hand and point it in the direction of the position vector (\(r\) or \(d\)), then turn your fingers in the direction of the force, and your thumb will point toward the direction of the torque.

1. If either the direction of \(r\) or the direction of \(F\) is reversed, the direction of  is reversed. If the direction of both \(r\) and \(F\) is reversed, there will be no change in the direction of \(\tau .\) The torque defined here is about a specific point, usually called the origin. The torque of the same force about a different origin is different. Hence, identifying the origin is very important.
2. If the force vector \(\theta = {0^{\rm{o}}}\) or \({180^{\rm{o}}},\) the force will not cause any rotation on the axis. It would either be shoving away from the axis of rotation because it is in the same direction or shoving towards the axis of rotation. The value of torque for these two cases is zero.
3. The most influential force vectors to produce torque are \(\theta = {90^{\rm{o}}}\) or \( – {90^{\rm{o}}},\) perpendicular to the position vector. This is because it will do the most to increase the rotation.

What Is the Unit of Torque?

Torque is represented by the Greek letter tau: \(\tau \) in calculations. When it is called the moment of force, it is represented by \(M.\) Torque is equal to the dot product between force and position vector. The SI unit of force is newton \((N),\) and the SI unit of distance is meter. Thus, 

\(\tau = \vec r \times \vec F = {\rm{meter – newton}}\)

Thus, the International System of Measurement units (SI units) used for torque is newton-meter or \( \rm{N} \times \rm{m}\).

Although the newton meter is equal to joules, which is the SI unit for energy or Work, Torque is measured in terms of newton-meter only.

In the imperial system, torque is measured in terms of pound-force-feet \(\left( {{\rm{lb – ft}}} \right){\rm{,}}\) which might be abbreviated as pound-foot, with the “force” implied.The dimensions of torque is \(\left[ {{M^1}{L^2}{T^{ – 2}}} \right].\)

Torque in Our Day-to-Day Life

Hinged Doors: The opening of a door and its hinges is caused by torque, and the hinges are the pivot point. If you try to open a door by pushing on the door near its hinges, it most likely will not open because there is not enough torque to force it to do so. When the door is pushed near the hinges, the length of the moment arm is not large enough to supply enough torque to open the door. To open the door, you have to push on the side of the door opposite the hinges to provide a substantial moment arm that allows for an increased torque to open the door.

Seesaws: Many people have had the experience of someone sitting on one end of the seesaw and another person sitting on the other end, and one person is heavier than the other. By sitting closer to the pivot, the heavier person can decrease their torque because the length of the moment arm will be shorter than that of the person lighter in weight. The smaller lever arm results in smaller torque allowing lighter people to lift heavier ones.

Wrenches: The nut (or bolt) is the rotation point because the user wants to tighten or loosen it by turning. The force is being exerted by the hand and arm. People use wrenches to exert a ninety-degree force on the nut or bolt.

Horsepower

We often hear people discussing horsepower and torque before buying a vehicle, especially when it comes to racing cars. A unit of measurement of power, Horsepower, is the rate at which work is done. We usually discuss horsepower for engines or motors.  Horsepower means the total power output of the engine. In very simple terms, if torque is the force you feel pushing you back in your seat on acceleration, then horsepower is the speed achieved at the end of that acceleration.

Mathematically, horsepower equals torque multiplied by rpm, divided by a constant. Now, there is generally a limit on how fast you can spin an engine of a vehicle, having higher torque allows for greater horsepower at lower RPMs (revolutions per minute).

Solved Problems on Torque

Q.1. A car mechanic applies a force of \({\rm{600}}\,{\rm{N}}\) to a wrench to loosen a bolt. He applies the force which is perpendicular to the arm of the wrench. The distance from the bolt to the mechanic’s hand is \({\rm{0}}{\rm{.40}}\,{\rm{m}}{\rm{.}}\) Find out the magnitude of the torque applied.
Ans: The angle between the moment the arm of the wrench and the force is without a doubt \({90^{\rm{o}}}\)
Thus, \(\sin \theta = {90^{\rm{o}}}\)
\(\theta = 1\)
The torque is: \(\tau = F \times r \times \sin \theta \)
Therefore, magnitude of the torque \( = (600\;{\rm{N}})(0.4\;{\rm{m}}) = 240\,{\rm{N}\,\rm{m}}\)
Hence, the magnitude of the torque is \(240\,{\rm{N}\,\rm{m}}.\)

Q.2. The washroom door is of width \({\rm{50}}\,{\rm{cm}}{\rm{.}}\) If the door handle is \({\rm{10}}\,{\rm{cm}}\) from the edge and the Force of \({\rm{2}}\,{\rm{N}}\) is applied on the handle. Compute the torque.
Ans: The handle of the door is located at \({\rm{10}}\,{\rm{cm}}{\rm{.}}\)
Distance between application of force and the edge of door \(d = 50\,- 5 = 45\;{\rm{cm}} = 0.45\;{\rm{m}}\)
Force exerted \( = 2\;{\rm{N}}\)
Torque \( = F \times d = 2\;{\rm{N}} \times 0.45\;{\rm{m}}\)
Torque \( = 0.9\,{\rm{Nm}}{\rm{.}}\)

Summary

Torque is the tendency of a force to cause or change the rotational motion of an object. Torque is also known as moment, or moment of force. In vector form, torque is the cross product of the radius vector (from the axis of rotation to the point of application of force) and the force vector. It is represented by the letter \({\vec \tau }\). Its formula is given as:

\(\vec \tau = \vec r \times \vec F\)

\(|\tau | = ||r|| \cdot ||F||\sin \theta \)

Where,  is the torque, \({\vec r}\) is the radius vector, \({\vec F}\) is the force vector and \(\theta \) is the angle between the force vector and the radius vector.

Torque is a vector quantity; it has both magnitudes as well as direction. To find the direction in which the torque acts on a given body, take your right hand and point it in the direction of the position vector (\(r\) or \(d\)), then turn your fingers in the direction of the force, and your thumb will point toward the direction of the torque.

The SI unit of torque is newton-meter or \( \rm{N} \times \rm{m}\) and the dimensions of torque is \(\left[ {{M^1}{L^2}{T^{ – 2}}} \right].\)

Torque is produced whenever there is rotation produced by the application of force and hence,  common examples of torque are wrenches or doors.

Frequently Asked Questions on Torque

Let’s look at some of the frequently asked questions about Torque:

Q.1. What is torque in a human body?
Ans: Torque is the driving force for human movement. Being able to manipulate the target muscle torque will allow for a more specific intervention. The moment arm of a force system is the perpendicular distance from an axis to the line of action of a force. Muscle torque is the force applied by the muscles through a moment arm of a given length at a given angle to the joint.

Q.2. What is torque in simple terms?
Ans: Torque is a measure of the force that can cause an object to rotate about an axis. Just as force causes an object to accelerate in linear kinematics, torque causes an object to acquire angular acceleration.

Q.3. Where is torque used?
Ans: The most obvious example of torque in action is the operation of a crescent wrench loosening a lug nut, a playground seesaw, a doorknob, opening a soda bottle, and even in-car engines and steering wheels.

Q.4. How is torque calculated?
Ans: Torque formula can be given as:
\(\tau = \vec r \times \vec F\)
The direction of the torque vector is found by convention using the right-hand rule.

Q.5. What is better hp or torque?
Ans: Torque is the ability of a vehicle to perform work — specifically, the twisting force applied by the crankshaft. Horsepower is how rapidly the vehicle can perform that work. Torque multiplied by rpm returns horsepower. As there is a limit on how fast you can spin an engine, having higher torque allows for greater horsepower at lower rpm.

Q.6. Does torque make a car faster?
Ans: Torque multiplied by rpm returns horsepower. Basically, the faster the crankshaft spins with the same amount of force, the more power an engine will make. A car with more hp than torque will always be quicker since this gives a car acceleration and speed.

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