Question

Let →F be a force acting on a particle having position vector →r. Let →Γ be the torque of this force about the origin, then
(a) →r.→Γ=0 and →F.→Γ=0
(b) →r.→Γ=0 but →F.→Γ≠0
(c) →r.→Γ≠0 but →F.→Γ=0
(d) →r.→Γ≠0 and →F.→Γ≠0

Answer 1

(a) →r.→Γ=0 and →F.→Γ=0   is suitable for the force →F acting on a particle with position vector →r and the torque is →Γ, about the origin.

Explanation:

Torque is Force Vector F cross product and Position Vector r cross product.

Thus T = r × F (vector form)

The resulting Vector (here T) is always perpendicular in a cross product to both of the two vectors whose product is found.

So, here T is both perpendicular to Forward and to r.

So, angle between T and F =90°

Similarly, angle between T and r=90°

The dot product is also for two vectors a and b

a.b = |a| |b| cos θ,    where θ is angle between a and b

Going on from the previous two sentences,

r.T = |r| |T| cos 90° = 0

F.T= |F| |T| cos 90° = 0

So, Option A, is a answer

r.T = 0 and F.T=0

The force →F acting on a particle having position vector →r and  →Γ be the torque of the force about the origin, then the relation between them is given by →r.→Γ=0 and →F.→Γ=0