Question

State whether true or false.
The SI unit of torque is Nm.
A. True
B. False

Answer 1

Hint: Describe force and try to connect it with torque and position vector.Then arrive at the equation of torque \[\mathbf{\tau }\text{ }=\text{ }\mathbf{r}\text{ }\times \mathbf{F}\]. Using this equation derives the unit of torque.

Complete Step-by-Step solution:
Basically, force is nothing but a push or a pull on an object. The force applied on a body, results in the change in the state of motion of that body in the linear direction. Which indicates that the body will accelerate linearly.
Whereas what happens if the force tends to rotate an object? Usually this happens in a three dimensional system. Torque is that effect which tends to rotate an object about its axis of rotation. So there will be a change in its angle. So we can conclude that torque results in the angular acceleration of an object about a fixed axis. It means that there will be a change in angular velocity.Or we can say that torque acts like an angular force.
The terms used forTorque or moment of force, rotational force or turning effect. The turning effect of force is called moment. It will rotate either in clockwise or anticlockwise direction.
The perpendicular distance between the object and the axis of rotation is called the position vector, r. There is an angle between this position vector and the line of force. This is represented as θ.Then the torque τ is defined as the product of the magnitude of force F and the position vector r. it is evident that torque is in the perpendicular direction of \[\mathbf{r}\times \mathbf{F}\]. Then
\[\mathbf{\tau }\text{ }=\text{ }\mathbf{r}\text{ }\times \mathbf{F}\]……………..(1)
or the cross product can be replaced by \[\sin \theta \overset{\wedge }{\mathop{n}}\,\], where \[\overset{\wedge }{\mathop{n}}\,\]is the unit vector in the direction τ.
Thus equation (1) can be written as
\[\overrightarrow{\tau }=\sin \theta \overset{\wedge }{\mathop{n}}\,\]………………(2)
Then the magnitude of the torque is given as
\[\left| \overrightarrow{\tau } \right|=\left| \right|\sin \theta \overset{\wedge }{\mathop{\left| n \right|}}\,\]………..(3)
Since \[\left| \overset{\wedge }{\mathop{n}}\, \right|=1\], above equation becomes
\[\left| \overrightarrow{\tau } \right|=\left| \right|\sin \theta \left| \right|\]…………….(4)
Here we can see torque completely depends on position vector, force and angle between r and F.
The dimensional of torque is [\[{{M}^{1}}{{L}^{2}}{{T}^{-2}}\]].
Since the SI unit of force is Newton(N) and distance is meter(m), the unit of torque will be Newton meter Nm.

Therefore the answer is true.

Note: We are not considering the unit of θ here. And remember in a given problem, the values of r and F won’t change. But the value of τ changes in accordance with θ. For example if \[\theta ={{90}^{0}}\], then \[r\mathbf{F}\]. If \[\theta =0\], τ=0. So the value of torque changes as the angle of rotation changes.