Question

9.) (a) If v= vi + v2, is 'v' necessarily greater than vi and/or v2? [Note that v, vị and v2 denote the magnitudes of vectors v, vi and v2 respectively]. (b) Two objects have the same momentum. Do the velocities of these objects necessarily have (a) the same directions and (b) the same magnitudes? (c) If a particle of mass 5 kg travels in uniform circular motion with a speed of 9.79 m/s, how much work is done by the centripetal force acting on it (a) at any instant and (b) during one complete revolution? Answer the question if the speed of the particle were 9.81 m/s.

Answer 1

9) a) From the equation it may seem that v is greater than v1 and v2 both.

But these are vector quantities and we cannot just assume the magnitude of them and add them up. They have directions too.

v1 and v2 may be directed opposite to each other for instance v1=5 m/s north and v2=6 m/s south

So, the resultant velocity will be (6-5) m/s south=1 m/s south which is less than both v1 and v2.

Again v1 and v2 can be equal to each other and directed opposite to each other. Then the resultant velocity v will be zero.

So the equation does not guarantee that v is greater v1 and/or v2

b) Momentum is a vector quantity and is given by the product of mass m and velocity v and is directed along the direction of velocity

p = m v

If both particles have same momentum i.e.,

立 = :

This indicates that the momentum vector is equal for both the objects i.e., both direction and magnitude.

a) Now for the direction of momentum to be same for both the masses their velocities must be directed along the same direction. So velocities of both the objects has the same direction.

b) Momentum is the product of mass and velocity. So for their magnitude of momentum to be equal the product of mass and velocity is equal.

The two particles can have different velocities but the product of mass and their velocities can be equal. So, velocity may not be equal in magnitude.

c) Centripetal force is directed along the radius of a circle towards the center which means that this force is always directed perpendicularly to the direction of displacement.

So, work done by the centripetal force F in displacing the object by a distance x is

W = F = Frcos A = Fr cos 90º = 0

So work done by the centripetal force in an uniform circular motion is always zero.

a) Work done is zero

b) Work done is zero