Question

The magnitude of torque on a particle of mass 1 kg is 2.5 Nm about the origin. If the force acting on it is 1 N, and the distance of the particle from the origin is 5 m, what is the angle between the force and the position vector? (in radians)
(A) $\dfrac{\pi }{8}$
(B) $\dfrac{\pi }{6}$
(C) $\dfrac{\pi }{4}$
(D) $\dfrac{\pi }{3}$

Answer 1

Hint:In order to solve this problem,we are going to apply the concept of torque.Torque acting on a body about a point is the cross product of the position vector and the force acting on the body. Its magnitude is given by the formula, $\tau = rF\sin \theta $.

Complete step by step answer:
The magnitude of torque of a body is given as 2.5 Nm. The mass of the body on which this torque is acting is 1 kg. The magnitude of the force acting on the body is 1 N. The body is at a distance of 5 m from the origin.
Torque is a result of the component of force perpendicular to the position vector acting on the body such that it does not pass through the axis of rotation of the body. It is expressed as,
$\vec \tau = \vec r \times \vec F$
The magnitude of the torque vector can be found by the product of the magnitudes of position vector and the force vector and the sine of the angle between position vector and force vector. It can be written as,
$\tau = rF\sin \theta $ …equation (1)
We need to find the angle between the position vector and the force vector. On substituting the values of torque, position, and force, we obtain,
$2.5 = 5 \times 1 \times \sin \theta \\
\Rightarrow \sin \theta = \dfrac{{2.5}}{5}\\
\therefore \sin \theta = \dfrac{1}{2}$
The value of sin $\theta $ is equal to $\dfrac{1}{2}$, when $\theta = {30^ \circ } = \dfrac{\pi }{6}$.Therefore, the angle between position vector and force is $\dfrac{\pi }{6}$.

Hence, the correct answer is option B.

Additional Information:
Torque due to a force F is more if the distance between the point of application of force and the point about which the body rotates is more. If the distance is more, a lesser amount of force is required to produce the same torque. This is why it is easier to open or close the door by applying force away from the hinge about which it rotates.

Note:Torque is a vector quantity given by the cross product of distance and force. The order in which distance and force are written in the formula for torque is important as the value may change on interchanging their positions, leading to an incorrect answer as the direction of torque is changed.