System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 34

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Weekly Quiz Competition

Question 1
1 / -0

The point through which the total weight of an object appears to act for any orientation of the object is ______.

A
Centre of buoyancy

B
Centre of gravity

C
Centre of curvature

D
Median of a triangle

Solution

The point through which the total weight of an object appears to act for any orientation of the object is centre of gravity.
Question 2
1 / -0

Two bodies of unequal masses possess the same momentum. Which of the following can be concluded?

A
Heavier mass possesses lesser K.E

B
The masses of the two bodies are in the direct ratio of their velocities

C
Lighter mass possesses less K.E,

D
Both (A) and (B)

Solution

Kinetic energy $$K = \dfrac{P^2}{2m}$$
For same momentum, $$K\propto \dfrac{1}{m}$$
Thus heavier mass possesses lesser kinetic energy.
Question 3
1 / -0

A particle is projected at an angle of $${30}^{0}$$ with the horizontal with a momentum $$P$$. At the highest point its momentum is

A
$$\dfrac { \sqrt { 3 } }{ 2} P$$

B
$$\dfrac { 2 }{ \sqrt { 3 } } P$$

C
$$P$$

D
$$\dfrac { 1 }{ 2 } P$$

Solution

At the point of projection, $$P=mu$$
At the highest point, vertical component of velocity is 0.
Therefore, $$u_x=ucos30^0$$

$$P'=m\left( u\cos { \theta } \right) $$

$$=mu\dfrac { \sqrt { 3 } }{ 2 } \\$$

$$=\dfrac { P\sqrt { 3 } }{ 2 } $$
Question 4
1 / -0

Two bodies of mass $$4 \ kg$$ and $$5\ kg$$ are moving along east and north directions with velocities $$5\ m/s$$ and $$3 \ m/s$$ respectively. Magnitude of the velocity of centre of mass of the system is:

A
$$\dfrac{25}{9} m/s$$

B
$$\dfrac{9}{25} m/s$$

C
$$\dfrac{41}{9} m/s$$

D
$$\dfrac{16}{9} m/s$$

Solution

Let east direction be x axis and north be y axis. Then, velocity of first body is $${ v }_{ 1 }=5i\quad m/s$$ and that of body 2 will be $${ v }_{ 2 }=3j\quad m/s$$. Since velocity of center of mass is given by

$${ v }_{ cm }=\dfrac { { m }_{ 1 }{ v }_{ 1 }+{ m }_{ 2 }{ v }_{ 2 } }{ { m }_{ 1 }+{ m }_{ 2 } } =\dfrac { 4(5i)+5(3j) }{ 4+5 } =\dfrac { 20 }{ 9 } i+\dfrac { 15 }{ 9 } j\quad m/s$$

Hence, magnitude of velocity is $$|{ v }_{ cm }|=\dfrac { \sqrt { 400+225 } }{ 9 } m/s=\dfrac { 25 }{ 9 } \ m/s$$.
Question 5
1 / -0

Which of the following quantities is zero, when a uniform object is being supported at its centre of gravity?

A
Mass

B
Weight

C
Force

D
Moment

Solution

Force of gravity on the object acts at its centre of gravity. If the uniform object is supported at its centre of gravity, net torque acting on it becomes zero.
Question 6
1 / -0

While opening a tap with two fingers, the forces applied by the fingers are:

A
equal in magnitude

B
parallel to each other

C
opposite in direction

D
all the above

Solution

A couple has to be applied to the tap in order to open it. A couple is the combination of two equal and opposite parallel forces acting at different axes. Thus option D is correct.
Question 7
1 / -0

Two particles $$A$$ and $$B$$ are connected by a rigid rod $$AB$$. The rod slides on perpendicular rails as shown in figure. The velocity of $$A$$ to the left is $$10m/s$$. What is velocity of $$B$$ when $$\alpha= 60^o$$? (in m/s)

A
$$4.3$$

B
$$5$$

C
$$8$$

D
$$17.3$$

Solution

Given : $$V_A = 10 m/s$$
Both the particles move can move parallel to the surface but not perpendicular to the surface.
From figure,
$$\tan 60^o = \dfrac{V_B}{V_A}$$

$$\implies V_B = V_A \tan 60^o $$
$$V_B = 10 \times \sqrt3$$
$$V_B = 17.3 \ \ m/s$$
Question 8
1 / -0

A sphere moving at some instant with horizontal velocity $$v_c $$ towards right and angular velocity in the anti-clockwise sense. If $$|v_c| = |\omega R|$$ The instantaneous center of rotation is:

A
at the bottom of the sphere

B
at the top of the sphere

C
at the centre of the sphere

D
Anywhere inside the sphere

Solution
Question 9
1 / -0

Which of the following has the maximum momentum?

A
A $$100 kg$$ vehicle moving at $$0.02 m/s$$

B
A $$4 g$$ weight moving at $$10000 cm/s$$

C
A $$200 g$$ weight moving with kinetic energy $$10^{6} J$$

D
A $$20 g$$ weight after falling $$1 km$$

Solution

For (A) : $$m =100 \ kg$$ and $$v = 0.02 \ m/s$$
Thus momentum $$P = mv = (100)(0.02) = 2 \ kg \ m/s$$
For (B) : $$m =4 \ g =0.004 \ kg$$ and $$v=10000\ cm/s = 100 \ m/s$$
Thus momentum $$P = mv = (0.004)(100) = .4 \ kg \ m/s$$
For (C) : Kinetic energy $$K = 10^6 \ J$$ and mass $$m = 0.2 \ kg$$
Thus momentum $$P = \sqrt{2mK} = \sqrt{2(0.2)(10^6)} = 632.45 \ kg \ m/s$$
For (D) : Mass $$m = 0.02 \ kg$$ and height $$h =1000 \ m$$
Thus kinetic energy acquired $$K= mgh = (0.02)(10)(1000) = 200 \ J$$
Thus momentum $$P = \sqrt{2mK} = \sqrt{2(0.02)(200)} = 2.83 \ kg \ m/s$$
Option C has the maximum momentum, so correct answer is option C.
Question 10
1 / -0

Two bodies have same mass and speed, then:

A
their momentums are same

B
the ratio of momentums is not determined

C
the ratio of their magnitudes of momentum is one

D
both (B) and (C) are correct

Solution

Given that the mass and speed of two bodies are same
We know that momentum is equal to $$p=mv$$
But the momentums of two bodies are not equal, because momentum is a vector quantity
The ratio of momentums is not defined because it is a vector quantity and ratio is a scalar quantity
The ratio of their magnitudes are same, Therefore option $$D$$ is correct

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System of Particles and Rotational Motion Test - 34

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System of Particles and Rotational Motion Test - 34

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Question 1

Centre of buoyancy

Centre of gravity

Centre of curvature

Median of a triangle

Solution

Question 2

Heavier mass possesses lesser K.E

The masses of the two bodies are in the direct ratio of their velocities

Lighter mass possesses less K.E,

Both (A) and (B)

Solution

Question 3

$$\dfrac { \sqrt { 3 } }{ 2} P$$

$$\dfrac { 2 }{ \sqrt { 3 } } P$$

$$P$$

$$\dfrac { 1 }{ 2 } P$$

Solution

Question 4

$$\dfrac{25}{9} m/s$$

$$\dfrac{9}{25} m/s$$

$$\dfrac{41}{9} m/s$$

$$\dfrac{16}{9} m/s$$

Solution

Question 5

Mass

Weight

Force

Moment

Solution

Question 6

equal in magnitude

parallel to each other

opposite in direction

all the above

Solution

Question 7

$$4.3$$

$$5$$

$$8$$

$$17.3$$

Solution

Question 8

at the bottom of the sphere

at the top of the sphere

at the centre of the sphere

Anywhere inside the sphere

Solution

Question 9

A $$100 kg$$ vehicle moving at $$0.02 m/s$$

A $$4 g$$ weight moving at $$10000 cm/s$$

A $$200 g$$ weight moving with kinetic energy $$10^{6} J$$

A $$20 g$$ weight after falling $$1 km$$

Solution

Question 10

their momentums are same

the ratio of momentums is not determined

the ratio of their magnitudes of momentum is one

both (B) and (C) are correct

Solution

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