Q38 of 42 Page 35

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


(a) Here, the x and y coordinates are

x = A cos ωt


y = B sin ωt


The equation of an ellipse is:



Substituting x and y, we find that LHS = RHS.


So the trajectory is an ellipse.


(b) To find force, we need to find acceleration.


This can be found by differentiating r two times with respect to t.



And




We know,




More from this chapter

All 42 →
36

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


 37

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.


 39

A cricket bowler releases the ball in two different ways

(a) giving it only horizontal velocity, and


(b) giving it horizontal velocity and a small downward velocity. The speed vs at the time of release is the same. Both are released at a height H from the ground. Which one will have greater speed when the ball hits the ground? Neglect air resistance.


 40

There are four forces acting at a point P produced by strings as shown in Fig. 5.11, which is at rest. Find the forces F1 and F2 .

.