If the external forces acting on a system have zero resultant, the centre of mass:

(a) must not move

(b) must not accelerate

(c) may move

(d) may accelerate

Choose the correct options:

1. (a) and (b)

2. (b) and (c)

3. (c) and (d)

4. All of these

A nonzero external force acts on a system of particles. The velocity and the acceleration of the centre of mass are found to be \(v_0\) and \(a_0\) at an instant \(t.\) It is possible that:
(a) \(v_0=0,\) \(a_0=0\)
(b) \(v_0=0,\) \(a_0 \neq0\)
(c) \(v_0 \neq0,\) \(a_0=0\)
(d) \(v_0 \neq0,\) \(a_0 \neq0\)
Choose the correct options:
1. (a) and (b)
2. (b) and (c)
3. (c) and (d)
4. (b) and (d)

Two balls are thrown simultaneously in the air. The acceleration of the centre of mass of the two balls while in the air:

Let \(\overrightarrow A\) be a unit vector along the axis of rotation of a purely rotating body and \(\overrightarrow B\) be a unit vector along the velocity of a particle P of the body away from the axis. The value of \(\overrightarrow A.\overrightarrow B\) is:
1. \(1\)
2. \(-1\)
3. \(0\)
4. None of these

A body is uniformly rotating about an axis fixed in an inertial frame of reference. Let \(\overrightarrow A\) be a unit vector along the axis of rotation and \(\overrightarrow B\) be the unit vector along the resultant force on a particle P of the body away from the axis. The value of \(\overrightarrow A.\overrightarrow B\) is:

1. 1

2. –1

3. 0

4. none of these

A particle moves with a constant velocity parallel to the X-axis. Its angular momentum with respect to the origin:

A body is in pure rotation. The linear speed \(v\) of a particle, the distance \(r\) of the particle from the axis and the angular velocity \(\omega\) of the body are related as \(w=\dfrac{v}{r}\). Thus:
1. \(w\propto\dfrac{1}{r}\)
2. \(w\propto\ r\)
3. \(w=0\)
4. \(w\) is independent of \(r\)

A body is rotating uniformly about a vertical axis fixed in an inertial frame. The resultant force on a particle of the body not on the axis is:

A body is rotating nonuniformly about a vertical axis fixed in an inertial frame. The resultant force on a particle of the body not on the axis is

1. vertical

2. horizontal and skew with the axis

3. horizontal and intersecting the axis

4. none of these

Let \(\vec{F}\) be a force acting on a particle having position vector \(\vec{r}\). Let \(\vec{\tau}\) be the torque of this force about the origin, then: