Question

Ramesh observes the motion of an insect in a circle. He finds that the insect travels 6revolutions in an anticlockwise direction for a time of 31.4s(a) Find the angular velocity of the insect.(b) If the insect travels 4 revolutions in the clockwise directions for a time of 8.6 sec,what will be the angular speed averaged over the total time? (c) Obtain the expression for centripetal acceleration in terms of angular speed

Answer 1

angular speed of the insect :  ω = angle / time duration
a) ω = 2π * 6 rad /31.4 sec = +1.2 rad/sec

b) ω = (4 * -2π rad) / 8.6 sec  = - 2.92 rad/sec

c)  centripetal acceleration = α = - r ω² = v² / r

See the diagram.

In  a circle the insect moves with a uniform speed | v |.

a = instantaneous acceleration = dv / dt

Let the center of circle be O. Let at t = t, the insect be at A (r , 0), where r = radius of the circle.   At time t = t +dt the insect moves to B by d Ф radians.  Let  dv be the change in the velocity from v to v+dv.  The angle between these two vectors is d Ф.

If we construct a triangle with vectors v and v+dv, then dv is the third side.  That is equal to v d Ф = length of the arc in the infinitesimally small triangle.  Magnitude of the vector v+dv = | v | = constant speed.

dv = v d Ф
dv/dt = v d Ф/dt = v ω = r ω² = v²/r  = a ,            as  v = r ω

The direction of  the centripetal acceleration is along the radius inwards.