System of Particles and Rotational Motion

Multiple Choice Questions (MCQs)

<math>\mathbf{Q.}\;\mathbf{1}</math> For which of the following does the centre of mass lie outside the body?

1. A pencil (b) A shotput (c) A dice (d) A bangle

Ans. (d) A bangle is in the form of a ring as shown in the adjacent diagram. The centre of mass lies at the centre, which is outside the body (boundary).

• C Centre

<math>\mathbf{Q.2}</math> Which of the following points is the likely position of the centre of mass of the system shown in figure?

Hollow sphere Air R/2

R/2 Sand

1. A (b) B (c) C (d) D

Thinking Process In a system of particles, the centre of mass of a body lies closer to heavier mass or masses.

Ans. (c) Centre of mass of a system lies towards the part of the system, having bigger mass. In the above diagram, lower part is heavier, hence CM of the system lies below the horizontal diameter.

110 NCERT Exemplar (Class XI) Solutions

<math>\mathbf{Q}.~\mathbf{3}</math> A particle of mass m is moving in yz-plane with a uniform velocity v with its trajectory running parallel to +ve y-axis and intersecting z-axis at <math>z = a</math> in figure. The change in its angular momentum about the origin as it bounces elastically from a wall at y = constant is

1. 2 mva <math>\hat{\mathbf{e}}_{x}</math> (c) <math>ymv \hat{\mathbf{e}}_x</math>

1. <math>mva \hat{\mathbf{e}}_{x}</math>

(d) 2 ym<math>v \hat{\mathbf{e}}_x</math>

Thinking Process In elastic collision, KE of the system remains conserved. Therefore, the ball will bounce back with the same speed v but in opposite direction i.e., along -ve y -axis.

Ans. (b) The initial velocity is <math>\mathbf{v}_i = \mathbf{v} \,\hat{\mathbf{e}}_y</math> and after reflection from the wall, the final velocity is <math>\mathbf{v}_{t} = -v\hat{\mathbf{e}}_{y}</math>. The trajectory is described as position vector <math>\mathbf{r} = y\hat{\mathbf{e}}_{y} + a\hat{\mathbf{e}}_{z}</math>. Hence, the change in angular momentum is <math>\mathbf{r} \times m(\mathbf{v}_f - \mathbf{v}_j) = 2mva\hat{\mathbf{e}}_x</math>.

<math>\boldsymbol{\bigstyle Q.}</math> <math>\boldsymbol{4}</math> When a disc rotates with uniform angular velocity, which of the following is not true?

1. The sense of rotation remains same
2. The orientation of the axis of rotation remains same
3. The speed of rotation is non-zero and remains same
4. The angular acceleration is non-zero and remains same

Ans. (d) We know that angular acceleration

<math>\alpha = \frac{d\omega}{dt}</math>, given <math>\omega = \text{constant}</math>

where <math>\omega</math> is angular velocity of the disc

<math>\alpha = \frac{d\omega}{dt} = \frac{0}{dt} = 0</math>

<math>\Rightarrow</math>

Hence, angular acceleration is zero.

<math>\mathbf{Q.5}</math> A uniform square plate has a small piece Q of an irregular shape removed and glued to the centre of the plate leaving a hole behind in figure. The moment of inertia about the z-axis is then,

Q o → hole

1. decreased

1. increased
2. the same (d) changed in unpredicted manner

• Thinking Process

For two bodies having same mass, the body having mass distributed at greater distance from an axis, will have more moment of inertia.