Torque and Equilibrium - JEE Important Topic

We are quite aware of what force is, both physically and theoretically. But have you experienced torque? Almost all of us have experienced or encountered torque in life. It is probably that we are not familiar with the term. So, what is torque? Remember playing on a see-saw? This is one of the best real-life examples of torque. Other instances are opening or closing hinged doors, opening a bottle cap and turning a steering wheel while driving a car. 

It is very easy at first to confuse torque with force because torque is a kind of force. However, while linear force causes or accelerates translational motion, torque is associated with rotational motion, the twisting or turning effect of an object. So, it can be said that torque creates a tendency to have a rotational motion in an object equivalent to the linear force causing translation of the same object.

Torque: Unit, Symbol, and Dimension

As explained above, torque is associated with rotational motion. Therefore, everything linear defining a translational motion caused by linear force has analogous quantities defining the rotational motion caused by torque. The acceleration in the rotational motion is known as angular acceleration, and the straight line around which rotation happens is known as the axis of symmetry. Consider a rod where two forces F₁ and F₂ are acting on its two ends in opposite directions.

In this case, the net force on the object will be zero if the magnitude of forces F₁ and F₂ directed opposite to each other are equal. So, there is no visible translational motion of the object. But if force is being applied, then there must be some effect. This effect is known as the rotation of the object about the centre or as a tendency to rotate. This is the turning effect of torque and is also known as the moment of force. 

Also, it has to be noted that the position of application of force is also important to achieve the rotation effect. It can be explained mathematically but think about it in general by taking the example of a hinged door. When you push a door very close to the hinged point, a greater amount of force is required to open the door. Whereas, pushing just at the end of the door where the knob is placed opens the door with very minimal effort.

Now, let us note down the parameters of Torque

• Unit of The Torque: 

SI Unit: N-m (kg-m²-s^-2)

C.G.S Unit: Dyne-Centimetre

• Symbol of Torque: $\tau$, M
• Dimension: $\left [M~{L^2}~{T^{-2}}\right]$

Equilibrium Force in Physics: Definition and Type

First, you should know what equilibrium is in physics. Equilibrium is a state or situation of any system where the net force equals zero, i.e., no matter what externally applied force tries to change the present state of the system, the system will not show any variation. This happens because there is an internal effect and cancellation of the effect on the system imposed externally, usually as opposed to each other.

So, mathematically, the conditions for equilibrium are

$\sum{F=0\ }$ ; 

This means the summation of force on the system is zero and is called force equilibrium. Now similarly, for torque equilibrium,

$\sum{\tau }=0$ 

So, when there is more than one torque acting on a system, then the summation of all those torques keeping in mind both the magnitude and direction, should be zero in torque equilibrium.

There are Three Types of Equilibrium in Physics:

• Stable Equilibrium: This is a state of a system when the body has the tendency to return to the original position after it has been subjected to any disturbance.

• Unstable Equilibrium: This state of system experiences large changes due to small perturbation in the system. The centre of gravity is displaced (lowered) and the body tends to move away from the original or mean position.

• Neutral Equilibrium: In this case, the centre of gravity remains at the suspension point when displaced by any force. 

How Do You Measure the Magnitude of Torque?

Let us learn about how to measure the magnitude of torque. From a simple observation mentioned above, we know that the distance or point of application of force from the axis of rotation is important to calculate the magnitude of the torque, and this distance is known as the lever arm.

Reaction torque is the kind of torque acting on a body which has restricting forces to overcome in order to rotate. The body when not in a state of free rotation develops reaction torque that eventually helps it to rotate.

Now, mathematically, torque can be expressed as

$\tau =F_{r}\sin \theta$ 

So, it can be concluded from the equation that the torque is dependent on the magnitude of the applied force F, the perpendicular distance between the point of application of force and the point through which the axis of rotation passes r (lever-arm length), and the angle  $\theta $ between the lever-arm and force vector as shown in the figure below. 

Now, let us understand the door opening example using the above equation. According to the above equation, when r is perpendicular to the force F then $\theta ={{90}^{\text{o}}}$ 

\[ \tau =Fr\sin \left( {{90}^{\text{o}}} \right) \] 

= \[ Fr \]

\[ \because \sin \left( {{90}^{\text{o}}} \right) =1 \] 

Now, why is it easier to open a door by pushing at a point away from the hinge? It’s because the value of r increases, and if the direction of the force is approximately perpendicular, then the force required will be very less in order to maintain the product. You can always apply more force, but a small amount of force will also do the job for you.

The Schematic Diagram of Torque

Torque is a vector quantity. So, it can be expressed in terms of a vector product as well. 

$\vec{\tau }=\vec{r}\times \vec{F}$ 

The cross product of force and position vector is equal to torque. 

How to Analyse the Direction of Torque?

Torque is a vector quantity, i.e., it has both magnitude and direction. It is very important to know the torque sign of application of force, especially when there are two or more competing forces causing the rotation. The best way to find the direction of the torque is by using the right-hand rule. According to this rule, the direction of the torque is perpendicular to both the force applied and the length of the lever arm as shown in the figure below. 

The torque is directed along the direction where angular acceleration changes with time. The steps to determine the torque using the right-hand rule are given below.

1. Use your right hand and hold it before you.

2. Curl all the fingers except the thumb with the rotation of the object. 

3. The direction towards which the thumb is pointing in the direction of the torque.

Conclusion

Torque is a force that causes turning action in systems. Torque is rotational equivalent to the linear force causing translation. The magnitude of torque depends on the force being applied, the length of the lever arm, and the angle between these two. The direction of torque can be determined via the right-hand rule. Torque equilibrium is achieved when the summation of all the torques acting on the system is zero.