Circular Motion & Rotational Dynamics

11. Torque

(a) The torque of force F about the point O is equal to the product of force and perpendicular distance of line of action of force from point.
τ = ¼ Force ½ x ¼ Perpendicular distance of line of action of force from point O)
= Fr sin θ = (F sin θ) r
= ¼ The component of force perpendicular to position vector) × ( Position vector )
∴ τ = Fr sin θ, r sin θ is known as lever arm
(b) Unit : In M.K.S = N-m and In C.G.S = dyne-cm
(c)dimension : M L² T^–2
(d) In vector form
= r F sin θ , where θ is angle between and and is unit vector perpendicular to the plane of nd .
(e) Torque is a vector quantity, whose direction is perpendicular to the plane of force and position vector and its direction is given by right hand screw rule.
(f) If the torque rotates the body in anticlockwise direction, the torque is positive and if the torque rotates the body in clock-wire direction, the torque will be negative.
(g) If a body is acted upon by more than one force, the total torque is the vector sum of each torque.

(h) τ = I α
I - Moment of inertia with respect to axis of rotation.
α - Angular acceleration with respect to axis of rotation
τ- Torque of force which is causing the rotational motion
(i) , where is angular momentum
(j) The more is the value of r, the more will be torque and easier to rotate the body.
i) The handle of screw driver is taken thick.
ii) In villages the handle of flour-mill is placed near the circumference.
iii) The handle of hand pump is kept-long.
iv) The r inch used for opening the tap, is kept-long.
(k) Work done by torque = = Torque × angular displacement.