Figure 5.9 shows (v_x ,t ),and( (v_y ,t) diagrams for a body of unit mass. Find the force as a function of time.

There are three forces F₁, F₂ and F₃ acting on a body, all acting on a point P on the body. The body is found to move with uniform speed.

(a) Show that the forces are coplanar.

(b) Show that the torque acting on the body about any point due to these three forces is zero.

When a body slides down from rest along a smooth inclined plane making an angle of 45° with the horizontal, it takes time T. When the same body slides down from rest along a rough inclined plane making the same angle and through the same distance, it is seen to take time pT, where p is some number greater than 1. Calculate the co-efficient of friction between the body and the rough plane.

A racing car travels on a track (without banking) ABCDEFA (Fig. 5.10). ABC is a circular arc of radius 2 R. CD and FA are straight paths of length R and DEF is a circular arc of radius R = 100 m. The co-efficient of friction on the road is μ = 0.1. The maximum speed of the car is 50 m s^-1. Find the minimum time for completing one round.

The displacement vector of a particle of mass m is given by r (t) = î A cos ωt + ĵ B sin ωt.

(a) Show that the trajectory is an ellipse.

(b) Show that F = −mω²r