Rotational equilibrium

Rotational motion is one of the most observed phenomena in the physical world. The rotating of the wheels, the fan spinning overhead, even the planet we live on is rotating about its axis. Therefore, rotational motion plays a very important part in kinematics.

According to thermodynamics, the entropy of the universe is constantly increasing. In a situation of increasing entropy, all bodies want to be at equilibrium. Let us have a look at what rotational equilibrium means for a rotating body and how it can be achieved.

Factors affecting rotational equilibrium

A body is put into rotational motion, when the angular acceleration is increased. This happens due to the application of a special type of force called torque or the moment of force. Along with torque, the moment of inertia also decides with how much acceleration a body will rotate. There are a few key factors that need to be adjusted in order to achieve rotational equilibrium.

1.Torque

Torque is described as the measure of any force that causes the rotation of an object about an axis. An intuitive feeling of torque can be experienced when a person opens the door to a room. When force is applied, the door rotates. The speed at which the door opens can be controlled by the amount of force that is applied, in other words, the torque applied on the door. 

Torque can be of two types: static and dynamic. When the torque on a body does not produce an angular acceleration, it is called static torque. On the other hand, when the torque on a body produces an angular acceleration, it is called dynamic torque.

2.Angular acceleration

When torque is applied to a rigid body already in rotation with a fixed angular velocity , then the application of the external torque results in the change in the angular velocity of the body. The measure of the change in angular velocity with respect to the time of a rigid body in rotational motion due to the application of an external torque is called angular acceleration. Let be the angular velocity of a rigid body, then the angular acceleration of the body is given as =d/dt.

3.Moment of inertia

When a rigid body is put into rotational motion, the amount of torque that will be required to change the angular velocity of the body is called its rotational inertia. Rotational inertia is a very important concept as it gives an idea of the torque required to achieve a certain objective. The rotational inertia of a body is also affected by the mass and also the distribution of the mass of the body with respect to the axis around which the body rotates. The distance of the centre of mass from the axis of rotation increases or decreases the rotational inertia of the rigid body.

The two conditions for rotational equilibrium

Now that we know the factors that affect rotational equilibrium, let us now have a closer look at what rotational equilibrium is and what the two conditions are to achieve it.

In a system under observation, when the observed parameters of the system do not change with time, the system is said to be in equilibrium. Extending this definition to rotational motion, when a rigid body is rotating, such that its parameters do not change with time, it is said to be in rotational equilibrium.

For example, when a rigid body is rotating with an angular velocity that does not change with its motion around a given axis. For a rigid body to be in rotational equilibrium, the total torque that is applied on the body should be zero.

The two conditions for achieving rotational equilibrium are:

1.Body is at rest and no torque is applied.

If a body is at rest, and no force is causing an angular displacement in the position of the body, then it is said to be in rotational equilibrium. In order to maintain rotational equilibrium in this stage, no external torque must be applied on the body. If an external torque is applied, then an equal and opposite torque must also be applied to keep the net torque on the body at zero.

2.Body is moving with a given angular velocity.

When an external torque is acting on a body but the torque is just enough to maintain a fixed angular velocity of the body, then the body is said to be in rotational equilibrium. The net torque on the body should not produce angular acceleration to ensure that the body stays in rotational equilibrium.

The formula for rotational equilibrium

Rotational equilibrium requires that the net torque acting on the body must be equal to zero to maintain rotational equilibrium. But torque is an effect of force. Therefore, the amount of force that is required to maintain net torque at zero can be found by adding all the torque on a body and equating it to zero.

We can visualise this with an example. Consider a flag hanging by a sideways post on the edge of a building. The weight of the flag will cause a force to act on the sideways post, and torque is produced. In order to negate the effect of this torque that will try to rotate the flag post, an external offsetting torque has to be applied. This is usually done by attaching a string from the end of the pole to the side of the edge. The tension on the string applies force on the rod in the opposite direction, inducing an opposite torque that will offset the torque produced due to the weight of the flag. Mathematically, if 1 is the torque due to the flag and 2 is the torque due to the tension on the string, then, to maintain rotational equilibrium,