Aparticle is to move in an xy plane, clockwise around the origin as seen from the positive side of the z axis. in unit vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is (a) 4.0 kg x m^2/s, (b) 4.0t^2 kg x m^2/s, (c) 4.0 sqrt(t) kg x m^2/s, and (d) 4.0/t^2 kg x m^2/s

a) τ =0,  b)  τ = 4.0 N m , c) τ = 2 / √t,  d)   τ = 2 t    N m

Explanation:

Angular momentum and torque are related by the equation.

       τ  =dL / dt

Bold indicates vectors, as time is a scalar, the direction of L is the same as the direction of the torque

Let's analyze each case presented

a) L = 4.0 kg m2 / s

As L is constant

    dL / dt = 0

    τ= 0

b) L = 4 t

     dL / dt = 4

     τ = 4.0 N m

c) L = 4 √ t

   dL / dt = 4 ½ 1/√t = 2 / √ t

   τ = 2 / √ t

d) L = 4 t²

   dL / dt = 4 (2t)

   τ = 2 t    N m