Question

For a rigid body, if the velocity of its centre of mass is zero at all times, then:

(i) Linear momentum of the body is zero
(ii) Net external force on the body must be zero
(iii) Kinetic energy of the body must be zero

Which of the above statement(s) is (are) correct?

Options:
a) (i)
b) (ii)
c) (i) and (ii)
d) (iii)

Answer 1

For a rigid body with a center of mass velocity of zero, the linear momentum is zero and the net external force must be zero, while kinetic energy could still be present due to rotational motion, making options (i) and (ii) correct.Option C is the correct answer.

Step-by-step explanation:

For a rigid body, if the velocity of its centre of mass is zero at all times, then the linear momentum, which is the product of the body's mass and velocity, is also zero. This is because linear momentum is directly proportional to the velocity of the body. Considering that the velocity of the center of mass is zero, it implies that there is no linear momentum; hence, statement (i) is correct.

Since Newton's first law states that a body at rest will remain at rest unless acted upon by a net external force, and we know the center of mass's velocity is zero (the body is at rest), this implies there is no net external force acting on the body, validating statement (ii).

However, a rigid body can have a zero velocity of the center of mass while still rotating around an axis. This rotation means that the body has rotational kinetic energy, so statement (iii) is not necessarily correct as kinetic energy could be present due to rotational motion.

Therefore, the correct answers are (i) and (ii), which correspond to option (c).