System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 36

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Question 1
1 / -0

Two particles A and B, initially at rest, moves towards each other under a mutual force of attraction. At the instant when the speed of A is $$u$$ and the speed of B is $$2 u$$, the speed of centre of mass is

A
Zero

B
$$u$$

C
$$1.5 u$$

D
$$3 u$$

Solution

As both the particles are at rest initially, thus centre of mass of the system is zero initially i.e. $${V_{cm}}_i = 0$$
Since the particles are moving towards each other under a mutual force of attraction i.e. no external force is acting on the system. Thus motion of centre of mass do not get affected and so velocity of centre of mass remains the same at every instant.
Hence velocity of centre of mass at the given instant is also zero i.e. $${V_{cm}}_f = {V_{cm}}_i = 0$$
Question 2
1 / -0

Two particles $$A$$ and $$B$$ (both initially at rest) start moving towards each other under a mutual force of attraction. At the instant when the speed of $$A$$ is $$v$$ and the speed of $$B$$ is $$2v$$, the speed of the centre of mass is

A
Zero

B
$$v$$

C
$$\cfrac { 3v }{ 2 } $$

D
$$-\cfrac { 3v }{ 2 } $$

Solution

Here $$ F_{ext} = 0 $$, hence momentum will remain constant.
$$p_i = p_f$$
$$ 0 = m_av_a + m_bv_b $$ ( $$p_i=0$$ since both were at rest initially)
$$ 0 = m_a \times v - m_b \times 2v $$

$$ m_b = \dfrac{m_a}{2} $$
Now $$V_{cm} = \dfrac {m_av_a + m_bv_b}{m_a + m_b}$$
Here $$v_a = v$$ and $$v_b= -2v$$

$$V_{cm}= \dfrac{m_a \times v - \dfrac{m_a}{2} \times 2v}{m_a+ \dfrac{m_a}{2}}$$
$$V_{cm}=0$$
Question 3
1 / -0

A calcite crystal is placed over a dot on a paper sheet and the crystal is rotated. On viewing through the calcite one sees

A
A single stationary dot

B
Two stationary dots

C
Two dots rotating about one another

D
One dot rotating about the other stationary dot-sometimes coinciding with it

Solution

This effect can also be observed by inking a dot on a piece of paper and placing a calcite crystal over it. One will observe two dots from above. If the crystal is rotated while kept flat on the paper, the dot produced by the e-ray will rotate around the stationary dot
Question 4
1 / -0

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

A
May move but not accelerate

B
May accelerate

C
Must not move

D
None of the above

Solution

Acceleration of center of mass $$a_{cm} = \dfrac{F_{external}}{M_{cm}}$$
From $$F_{external}=0$$, we get $$a_{cm}=0$$
Thus the acceleration of center of mass must be zero when no external force is acting on the system.
So, the center of mass either can be at rest or move with constant velocity but it must not accelerate.
Question 5
1 / -0

The centre of gravity of a rod (of length $$L$$), whose linear mass density varies as the square of the distance from one end is at :

A
$$\dfrac { 3L }{ 5 } $$

B
$$\dfrac { 2L }{ 5 } $$

C
$$\dfrac { L }{ 3 } $$

D
$$\dfrac { 3L }{ 4 } $$

Solution

Le the linear density be $$\rho=kx^{2}$$
Centre of Gravity$$=\dfrac{\int x\rho}{\int\rho}=\dfrac{\int_{0}^{L}x^{3}dx}{\int_{0}^{L}x^{2}dx}=\dfrac{3L}{4}$$
So, Option (D)
Question 6
1 / -0

The motion in which all points of a moving body move uniformly in the same line or direction is known as

A
Rolling

B
Rotatory

C
Both A and B

D
Translational

Solution

If a body rotates its motion its rotatory. Rolling motion is combination of rotation and translation motion.
Only in translational motion body moves along a straight line, that means body doesn't rotate then its motion is translational.
Question 7
1 / -0

A diatomic molecule is formed by two atoms which may be treated as mass points $$m_1$$, and $$m_2$$ joined by a massless rod of length r. Then, the moment of inertia of the molecule about an axis passing through the centre of mass and perpendicular to rod is :

A
Zero

B
$$(m_1+m_2)r^2$$

C
$$\dfrac{(m_1+m_2)}{m_1m_2}r^2$$

D
$$\left ( \dfrac{m_1m_2}{m_1+m_2} \right )r^2$$

Solution

Let point C be the centre of mass situated at distance x from atom of mass mi and at distance (r - x) from mass $$m_2$$. By definition of centre of mass, we have
$$m_1x= m_2(r - x)$$
$$ x =\dfrac{m_2}{ (m_1 + m_2)}r$$
Moment of inertia about the centre of mass is
$$I=m_1x^2+m_2(r-x)^2$$
or $$I=m_1\left ( \dfrac{m_2r}{m_1+m_2} \right )^2+m_2\left ( r-\dfrac{m_2r}{m_1+m_2} \right )^2$$
$$m_1 \left ( \dfrac{m_2r}{m_1+m_2} \right )^2+m_2\left ( \dfrac{m_1r}{m_1+m_2} \right )^2$$

$$\dfrac{m_1m_2}{m_1+m_2}r^2$$
Question 8
1 / -0

Sagar was playing with the spinning top, after some time he realized that motion of top is same as motion of earth spinning about its own axis. Identify the type of motion?

A
Rotational motion

B
Oscillatory motion

C
Rectilinear motion

D
None of these

Solution

Motion of earth spinning about its own axis is a rotational motion, thus the motion of spinning top is also a rotational motion.
Question 9
1 / -0

Which of the following doesn't represent rotatory motion?

A

B

C

D
Both A and C

Solution

In picture A, the running people are exhibiting translational motion. In picture B, the wheel rotates about a fixed axis and thus it exhibits rotatory motion whereas in picture C, the bob of pendulum oscillates to and fro about its mean position and thus it exhibits oscillatory motion.
Hence options A and C are correct.
Question 10
1 / -0

A body of mass $$m_{1} = 4\ kg$$ moves at $$5\hat {i}m/s$$ and another body of mass $$m_{2} = 2\ kg$$ moves at $$10\hat {i} m/s$$. The kinetic energy of centre of mass is :

A
$$\dfrac {200}{3}J$$

B
$$\dfrac {500}{3}J$$

C
$$\dfrac {400}{3}J$$

D
$$\dfrac {800}{3}J$$

Solution

$$v_{CM} = \dfrac {m_{1}\dfrac {dr_{1}}{dt} + m_{2} \dfrac {dr_{2}}{dt}}{m_{1} + m_{2}}$$
$$= \dfrac {4\times 5\hat {i} + 2\times 10\hat {i}}{4 + 2}$$
$$v_{CM} = \dfrac {40\hat {i}}{6} = \dfrac {20}{3}\hat {i}$$
The kinetic energy
$$K = \dfrac {1}{2}mv^{2}$$
$$= \dfrac {1}{2}\times (4 + 2) \times \dfrac {20\times 20}{3\times 3}$$
$$= \dfrac {1}{2}\times 6\times \dfrac {20\times 20}{3\times 3}$$
$$K = \dfrac {400}{3}J$$.

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System of Particles and Rotational Motion Test - 36

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System of Particles and Rotational Motion Test - 36

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Question 1

Zero

$$u$$

$$1.5 u$$

$$3 u$$

Solution

Question 2

Zero

$$v$$

$$\cfrac { 3v }{ 2 } $$

$$-\cfrac { 3v }{ 2 } $$

Solution

Question 3

A single stationary dot

Two stationary dots

Two dots rotating about one another

One dot rotating about the other stationary dot-sometimes coinciding with it

Solution

Question 4

May move but not accelerate

May accelerate

Must not move

None of the above

Solution

Question 5

$$\dfrac { 3L }{ 5 } $$

$$\dfrac { 2L }{ 5 } $$

$$\dfrac { L }{ 3 } $$

$$\dfrac { 3L }{ 4 } $$

Solution

Question 6

Rolling

Rotatory

Both A and B

Translational

Solution

Question 7

Zero

$$(m_1+m_2)r^2$$

$$\dfrac{(m_1+m_2)}{m_1m_2}r^2$$

$$\left ( \dfrac{m_1m_2}{m_1+m_2} \right )r^2$$

Solution

Question 8

Rotational motion

Oscillatory motion

Rectilinear motion

None of these

Solution

Question 9

Both A and C

Solution

Question 10

$$\dfrac {200}{3}J$$

$$\dfrac {500}{3}J$$

$$\dfrac {400}{3}J$$

$$\dfrac {800}{3}J$$

Solution

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