Torque and other basic concepts

Torque

Have you ever had a tough time unscrewing the cap off of something? Maybe a bottle or even a jar? Perhaps you were trying to open a jar of pickles or olives, and you used way too much force, and the resulting torque screwed the lid tighter. Just when you think you are going to break the darn cap open to get the contents of that jar, it starts turning smoothly until it eventually comes right off. This force is exactly what torque is.

Torque is a measure of the tendency of a force to rotate an object about an axis, fulcrum, or pivot. As a force is a push or a pull on an object, a torque can be a twist on the object. It is denoted by τ. The torque formula is 

Τ = r F

Τ = |r| |F| sinQ  

Τ = Torque vector

r = position vector about which the torque is measured

F = Force vector 

Q = Angle between the force vector and level arm vector

A force applied at a right angle to a lever multiplied by the distance from its point of application to the lever’s pivotal point typically produces a torque. The SI unit of torque is the newton metre(N-m).

Dimensional Analysis 

The dimensional formula of torque is [ML2T−2] as the torque formula is derived from force and position. For force, the dimensions are [MLT-2], and the dimensions are [L] for position vectors. So, calculating them, we get [ML2T−2].

Measurement of Torque 

For the explanation and measurement of torque, let’s look at the figure given below and see how the forces act on any object.

• Suppose there is a rod with endpoints A and B. Let two opposite forces, F1 and F2,  act on the given rod in a circular manner. However, the magnitude of both forces are equal, so the object has been balanced. So, we can conclude that the net force is zero.
• But somehow, the rod is rotating, and hence we can assume that even after being in translational equilibrium, the rod still can rotate because of the factor we call torque or moment of force.
• Here the most important thing to note is that torque is always perpendicular to the force acting on it. 
• Apart from it, we can see some other examples of how torque is calculated. When someone opens a door, he is applying force on the door, and the position vector will be perpendicular to it. 
• Therefore, we take a cross product between Force and the position vector, which comes into play as sinQ.

So, we see that torque is a vector. This means that it has both magnitude and direction. 

Torque Formula 

Torque formula depends on the angular momentum as for a single point particle, we can write it as :

L = r p

Here, “p” is the particle’s linear momentum, and “r” is the particle’s position vector. After this, the time derivative will be :

dL/dt = r dp/dt + dr/dt p

Now with the product rule in vectors, the result can be obtained. Besides this, we will use the previous results like dp/dt = F and dr/dt = v. So we can write it as :

dL/dt = (r F) + (v p)

When we do the cross-product between velocity and momentum, we will get zero because both the terms are parallel to get cancelled. So, now only two terms will be left, and they will be Force and position vector.

dL/dt = r Fnet = Τnet

The above proof can be generalised to any distribution of masses numerically. For example, if you have a continuous mass of points, and each point has a mass m, the static equilibrium is provided by the sum of forces over each point.

Types of Torque 

Torque is categorised into two types and is divided based on angular acceleration. These types are :

                 1. Static 

Static torque is a mechanism to study the form and magnitude of torque without considering the angular acceleration of an object which may be in motion. A particular case is a force applied at the end of a lever that acts to rotate the lever but does not cause it to accelerate. Some other examples of this are:

• When a force is applied on a closed door, it is static torque as the door isn’t moving and does not carry any acceleration.
• If a cyclist is pedalling at a constant speed, it is also static torque as there is no acceleration.

                    2. Dynamic

Dynamic torque comes into play when a body is producing angular acceleration from the start and, hence, moves along the given line or a particular track in it. For example:

• The drive shaft in a racing car, accelerating from the start line, is experiencing dynamic torque—the greater the acceleration of the racing car, the greater the torque and vice versa. 

Conclusion

This article went through torque, measurement of torque, and torque’s formula. Besides this, we also got to know about the types of torque, how torque comes into play in our daily lives, and its applications.