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Relation Between Torque and Speed

Angular Velocity

Angular velocity is a quantity defined in physics that describes the rate at which any object is rotating around an axis. It is also called the angular velocity vector because it is a vector quantity, and its direction is defined instantaneously. Angular velocity is the angular equivalent of the translational quantity velocity. In the same way, the translational velocity of an object is the measure of how fast an object is changing its coordinate in space and time. Angular velocity tells us at what rate the angle of an object is changing with respect to a fixed axis.

Rotational velocity, despite being constituted of translational motion in its essence, is quite a different quantity, and it plays a very important role in the study of the behaviour of subatomic particles, which have a very well defined rotational behaviour without having any real mass or size. So, in the case of subatomic particles, the angular motion takes on a very distinct aspect, which is entirely separate from the translational motion.

Angular velocity can be of two types: orbital and spin. Orbital angular velocity describes the motion of an object with respect to an axis that lies outside the body of the object. This means that the object revolves around a foreign point and all the points that lie in the body of the object move in the same direction at all times. A simple example of this kind of motion is the motion of the earth around the sun. Sun is a foreign object around which the earth revolves.

The other kind of angular velocity, spin angular velocity, refers to the motion of an object about an axis that lies in the body of the object. This means that the axis around which the object is revolving passes through the body of the object. In this case, different points on the object may be travelling in the opposite direction at the same time, as is the case with the pair of points that lie on the opposite side of the axis. The spin angular velocity is independent of the choice of origin, and it does not change if a different coordinate system is used.

Angular velocity is measured in terms of angle per unit time. Since angle has no physically defined dimension associated with it, the physical dimension of the angular velocity is time inverse. The SI unit of angular velocity is radians per second, and the symbol that is used to denote angular velocity is ω. When a body rotates in the clockwise direction, its angular velocity is positive, and if the body rotates in a counter-clockwise direction, its angular velocity is negative.

Torque

Torque is the angular equivalent of force in the translational scheme of things. So torque is responsible for providing the impetus for a rotating body to rotate. For different people studying, torque can assume different names like moment, a moment of force, rotational force, turning effect etc.

The concept of torque goes back to the time of Archimedes when he was trying to study levers and their motion. He figured that if linear force affects an object by displacing it from one point to another, a ‘twisting force’ must exist that causes an object to rotate on its axis.

Torque is defined as the vector product of linear instantaneous force and the distance of that force with the axis around which it is rotating. Different points on the same body may experience different torque. However, if the body is point-size, then the torque on all of the body is the same. The symbol τ generally denotes torque. When torque is being studied as the moment of force, it is referred to as M.

Relationship Between ω and τ

Torque, being the rotational equivalent of linear force, performs work when it is applied to any body. This work has the same dimensions as the linear work and is also expressed in the same units as the linear work, that is, joules.

The amount of work done by a rotating body can be found by using the relationship between torque and angular velocity. This relationship is given below:

P = τ⋅ω

Here,

P is the power generated

τ is the torque being applied to the body

ω is the angular velocity of the body

Since power is the work done per unit time, we can easily calculate work done by multiplying power by the time period for which the power is supplied.

 

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