System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 82

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Question 1
1 / -0

The linear mass density, $$\lambda (x)$$, for a one-dimensional object of length $$2{ x }_{ 0 }$$ is plotted in the graph. The location of the center of mass for this object, from origin is $$r_0$$ then $$\frac { 9{ r }_{ 0 } }{ { x }_{ 0 } } $$ is _____

A
11

B
15

C
18

D
25

Solution
Question 2
1 / -0

If $$I_2$$ is the moment of inertia of a thin rod about an axis perpendicular to its length and passing through its centre of mass and $$I_2$$ is the moment of inertia of the ring formed by the same rod about an axis tangent to the ring and perpendicular to the plane of the ring, then find the ratio $$I_1/I_2$$

A
$$\cfrac {\pi^2} 3$$

B
$$\cfrac {\pi^2} 6$$

C
$$\cfrac {\pi^2} 2$$

D
$$\cfrac {\pi^2} 5$$

Solution
Question 3
1 / -0

An object comprises of a uniform ring of radius $$ R $$ and its uniform chord AB (not necessarily made of the same material) as shown. Which of the following can not be the center of mass of the object?

A
$$ \left( \frac { R } { 13 } , \frac { R } { 13 } \right) $$

B
$$ \left( \frac { R } { \sqrt { 2 } } , \frac { R } { \sqrt { 2 } } \right) $$

C
$$ \left( \frac { R } { 4 } , \frac { R } { 4 } \right) $$

D
None of these

Solution
Question 4
1 / -0

The radius of gyration of a homogeneous solid cylinder, of length $$L$$ and radius $$R$$, for rotation about an axis perpendicular to its length through its one end is

A
$$\cfrac{L}{\sqrt{3}}$$

B
$$\sqrt{\cfrac{3}{2}}R$$

C
$$\cfrac { 1 }{ 2\sqrt { 3 } } { \left( 3{ R }^{ 2 }+4{ L }^{ 2 } \right) }^{ \cfrac { 1 }{ 2 } }$$

D
$$\sqrt { \cfrac { { R }^{ 2 } }{ 4 } +\cfrac { { L }^{ 2 } }{ 12 } } $$

Solution
Question 5
1 / -0

A uniformly straight rod is placed in vertical position on a smooth horizontal surface and released. As the rod is in motion, the center of mass moves

A
Horizontally

B
Vertically down

C
In a parabolic path

D
Does not move
Question 6
1 / -0

A 2 kg body and a 3 kg body are moving along the x-axis. At a particular instant the 2 kg body has a velocity of 3 $${ ms }^{ -1 }$$ and the 3 kg body has the velocity of 2 $${ ms }^{ -1 }$$. The velocity of the centre of mass of that instant is

A
5 $${ ms }^{ -1 }$$

B
1 $${ ms }^{ -1 }$$

C
0

D
None of these

Solution
Question 7
1 / -0

Two particles of masses m1 and m2 separated by a distance d are at rest initially. If they move towards each other under mutual interaction, where will they meet ?

A
at the centre of line joining the two particles

B
anywhere in between two masses

C
at the centre of mass of the system of two particles

D
none of the above

Solution
Question 8
1 / -0

The ratio of the radii of gyration of a circular disc and a circular ring of thr same radii about a tangential axis in the plane is:

A
$$1:2$$

B
$$\sqrt 5 : \sqrt 6$$

C
$$2:3$$

D
$$2:1$$

Solution

Let the radii of disc and ring is $$R$$
For Disc,
Moment of inertia of disc about diameter axis is $$\dfrac{M{ R }^{ 2 }}{4}$$
Radius of gyration of circular disc is
$$I=I_{cm}+MR^2$$

$$I=\dfrac { M{ R }^{ 2 } }{ 4 } +M{ R }^{ 2 }\\ \quad =M{ R }^{ 2 }(\dfrac { 1 }{ 4 } +1)\\ \quad =\dfrac { 5 }{ 4 } M{ R }^{ 2 }=M{ K }^{ 2 }\\ { K }^{ 2 }=\dfrac { 5 }{ 4 } { R }^{ 2 }\\ K=\dfrac { \sqrt { 5 } R }{ 2 }$$

For RIng
Moment of inertia of Ring about diameter axis is $$\dfrac{MR^2}{2}$$
Radius of gyration of circular ring
$$I=\dfrac { M{ R }^{ 2 } }{ 2 } +M{ R }^{ 2 }\\ \quad =M{ R }^{ 2 }(\dfrac { 1 }{ 2 } +1)\\ \quad =\dfrac { 3 }{ 2 } M{ R }^{ 2 }=M{ K }^{ 2 }\\ { K }^{ 2 }=\dfrac { 3 }{ 2 } { R }^{ 2 }\\ K=\dfrac { \sqrt { 3 } R }{ \sqrt { 2 } } $$

Ratio of radius of gyration
$$\dfrac { { K }_{ disc } }{ { K }_{ ring } } =\dfrac { \sqrt { 5 } R }{ 2 } \times \frac { \sqrt { 2 } }{ \sqrt { 3 } R } \\ \quad \quad \quad \quad \quad =\dfrac { \sqrt { 5 } }{ \sqrt { 3\times 2 } } \\ \quad \quad \quad \quad \quad =\dfrac { \sqrt { 5 } }{ \sqrt { 6 } } $$

Correct option is B.
Question 9
1 / -0

A shell is fired from a cannon with a velocity V at an angle $$\theta$$ with the horizontal direction. At the highest point in its path, it explodes into two pieces of equal masses. One of the pieces retraces its path to the cannon. The speed of the other piece immediately after the explosion is.

A
$$3 V \, cos \theta $$

B
$$2 V \, cos \theta $$

C
$$\frac {3}{2} V \, cos \theta $$

D
$$V \, cos \theta $$

Solution
Question 10
1 / -0

Dozen balls kept in a box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass.

A
of the box remains constant

B
of the box plus the ball system remains constant

C
of the ball remains constant

D
of the ball relative to the box remains constant.

Solution

No external force is acting on the (ball+box) system when the ball collides with the wall of the box i.e $$F_{ext}=0$$
Thus acceleration of centre of mass of the system
$$a_{cm} = \dfrac{F_{ext}}{ M} = 0$$
$$\Rightarrow \Delta v=0$$
Thus the velocity of the (ball+box) system remains constant.