System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 53

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Question 1
1 / -0

A disc of mass $$10kg$$ has moment of inertia of $$90kg$$ $${m}^{2}$$ about the axis which lies in the plane of disc and passing through its centre. The radius of gyration of disc about this axis is

A
$$3m$$

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

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

D
$$\sqrt{\cfrac{3}{2}}m$$

Solution
Question 2
1 / -0

Three identical masses are kept at the corners of an equilateral triangle ABC. A moves towards B with a velocity V, B moves towards C with velocity V, and C moves towards A with same velocity V. Then the velocity of center of mass of the system of particles is

A
V

B
zero

C
3V

D
$$\dfrac { V }{ 3 } $$

Solution
Question 3
1 / -0

The moment of inertia of two equal masses each of mass m at separation l connected by a rod of mass M. about an axis passing through centre and perpendicular to length of rod is.

A
$$\dfrac{{\left( {M + 3m} \right){L^2}}}{{12}}$$

B
$$\dfrac{{\left( {M + 6m} \right){L^2}}}{{12}}$$

C
$$\dfrac{{M{L^2}}}{4}$$

D
$$\dfrac{{M{L^2}}}{12}$$

Solution
Question 4
1 / -0

Find the gravitational force of attraction between a particle of mass m and a uniform slender rod of mass M and length L for the two orientations shown in the figure.

A

B

C
none of these

D
both A and B.
Question 5
1 / -0

A football rolls through the ground. The path followed by center of mass of football is:

A
linear

B
circular

C
rotational

D
all the above

Solution

When a football rolls through the ground, the path followed by center of mass of ball is linear as the center of mass remains always at a fixed height from the ground when the football rolls and moves in a straight line if the ball does not changes its direction of rolling which is assumed in this case.
The correct option is A.
Question 6
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 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
$$\left( \cfrac { { m }_{ 1 }+{ m }_{ 2 } }{ { m }_{ 1 }{ m }_{ 2 } } \right) {r}^{2}$$

D
$$\left( \cfrac { { m }_{ 1 }{ m }_{ 2 } }{ { m }_{ 1 }+{ m }_{ 2 } } \right) {r}^{2}$$

Solution
Question 7
1 / -0

Centre of mass of two uniform rods of the same length but made up of different materials and kept as shown, if the meeting point is the origin of co-ordinates

A
$$(L/4),\, (L/4)$$

B
$$(2L/3),\, (L/2)$$

C
$$(L/3),\, (L/3)$$

D
$$(L/3),\, (L/6)$$

Solution
Question 8
1 / -0

Centre of mass is a point

A
Which is geometric centre of a body

B
From which distance of particles are same

C
Where the whole mass of the body is supposed the

D
none of these

Solution

Center of mass of a body is a point where we can assume all mass is concentrated.
Question 9
1 / -0

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

A
$$( \mathrm { R } / 3 , \mathrm { R } / 3 )$$

B
$$( R / 2 , R / 2 )$$

C
$$( \mathrm { R } / 4 , \mathrm { R } / 4 )$$

D
none of the above

Solution

$$According\, to\, question\\ the\, chord\, is\, a\, uniform\, \, chord\, and\, the\, center\, of\, mass\, coordinate\, \, of\, \, system\, will\, be\, \\ displaced\, in\, x\, and\, y\, coordinate\, \, by\, same\, value.\, \\ Hence,\, \, X\, and\, Y\, coordinate\, will\, be\, same. \\ so\, that\, wrong\, opton\, is\, D.\, \, (None\, of\, the\, above)$$
Question 10
1 / -0

The object in figure is constructed of uniform wire bent into the shape from for the figure shown below the center of mass w.r.t. given frame is

A
$$\left( \frac { 1 } { 2 } \cos \frac { \theta } { 2 } , \frac { 1 } { 2 } \sin \frac { \theta } { 2 } \right)$$

B
$$\left( \frac { 1 } { 4 } \cos ^ { 2 } \left( \frac { \theta } { 2 } \right) \cdot \frac { 1 } { 8 } \sin \theta \right)$$

C
$$\left( \frac { 1 } { 2 } \sin ^ { 2 } \left( \frac { \theta } { 2 } \right) , \frac { 1 } { 2 } \right)$$

D
$$\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \cos ^ { 2 } \theta \right)$$

Solution

$$\begin{array}{l} Ycom=\dfrac { { { m_{ 1 } }{ y_{ 1 } }+{ m_{ 2 } }{ y_{ 2 } } } }{ { { m_{ 1 } }+{ m_{ 2 } } } } \\ =\dfrac { { { m_{ 1 } }\left( 0 \right) +m\left( { \dfrac { l }{ u } \sin \theta } \right) } }{ { 2m } } \\ =\dfrac { l }{ 8 } \sin \theta \\ Hence,\, the\, option\, B\, is\, the\, correct\, answer. \end{array}$$