System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 78

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Question 1
1 / -0

Six identical particles each of mass 'm' are arranged at the corners of a regular hexagon of side length "L". If the mass of one of the particle is doubled, the shift in the center of mass is

A
$$L$$

B
$$6L/7$$

C
$$L/7$$

D
$$\cfrac{L}{\sqrt{3}}$$
Question 2
1 / -0

Four masses m, 2m, 3m, & 4m, are kept on a straight line at 0, a, 2a & 3a as shown in figure. The radius of gyration of the system about axis AB is

A
$$\sqrt { 2 } a$$

B
$$\sqrt { 3 } a$$

C
2 a

D
$$\sqrt { 5 } a$$

Solution
Question 3
1 / -0

The position of axis of rotation of a body is changed so that its moment of inertia decreases by 36$$\%$$ . The $$\%$$ change in itsradius of gyration is

A
decreases by 18$$\%$$

B
increases by 18$$\%$$

C
decreases by 20$$\%$$

D
increases by 20$$\%$$

Solution
Question 4
1 / -0

Two objects initially some distance apart are released from rest. They move towards each other under mutual gravitational force. At some instant, speed of one of them is $$v$$ and that of other is $$v / 4$$ .Then speed of their centre of mass at this instant is:

A
$$\cfrac { 3 v } { 4 }$$

B
$$\cfrac { v } { 4 }$$

C
$$\cfrac { 5 v } { 4 }$$

D
zero
Question 5
1 / -0

A solid sphere rolls down an inclined plane and its velocity at the bottom is $${ v }_{ 1 }$$. The same sphere slides down the plane (without friction) and its velocity at the bottom is $${ v }_{2 }$$. Which of the relations given below is correct?

A
$${ v }_{ 1 }={ v }_{ 2 }$$

B
$${ v }_{ 1 }=\sqrt { \frac { 5 }{ 7 } } { v }_{ 2 }$$

C
$${ v }_{ 1 }=\sqrt { \frac { 7 }{ 5 } } { v }_{ 2 }$$

D
None of these

Solution
Question 6
1 / -0

Find the moment of inertia of three identical rods each of mass $$m$$ and length $$l$$ forming an equilateral triangle, about one of its sides

A
$$\dfrac {ml^2}{4}$$

B
$$\dfrac {ml^2}{2}$$

C
$$\dfrac {3ml^2}{4}$$

D
$$\dfrac {2ml^2}{3}$$

Solution
Question 7
1 / -0

Half of the rectangular plate shown in figure is made of a material of density $$\rho _ { 1 }$$ and the other half of density $$\rho _ { 2 }$$.The length of the plate is $$L$$. Locate the centre of mass of the plate.

A
$$\cfrac { \left( \rho _ { 1 } + 3 \rho _ { 2 } \right) L } { 4 \left( \rho _ { 1 } + \rho _ { 2 } \right) }$$

B
$$\cfrac { \left( \rho _ { 1 } + 3 \rho _ { 2 } \right) L } { 2 \left( \rho _ { 1 } + \rho _ { 2 } \right) }$$

C
$$\cfrac { \left( 3 \rho_ { 1 } + \rho _ { 2 } \right) L } { 2 \left( \rho _ { 1 } + \rho _ { 2 } \right) }$$

D
none

Solution
Question 8
1 / -0

Three identical rods each of mass m and length $$\ell$$ are joined to form an equilateral triangle. The moment of inertia about an axis passing through one corner and perpendicular to the plane of triangle, is

A
$$\frac { 4 } { 3 } m \ell ^ { 2 }$$

B
$$\frac { 9 } { 8 } m \ell ^ { 2 }$$

C
$$\frac { 3 } { 4 } m \ell ^ { 2 }$$

D
$$\frac { 3 } { 2 } m \ell ^ { 2 }$$

Solution

$$\begin{array}{l}\text { Height of equilateral triangle }=\frac{\sqrt{3}}{2} l \\\text { Moment of inertia at end of } \mathrm{rod}=\frac{\mathrm{ml}^{2}}{3} \\\text { and at corner is }=\frac{\mathrm{ml}^{2}}{12}\end{array} $$

$$\begin{aligned} I_{n 0} t &=I_{1}+I_{2}+I_{3}+m\times\left(\frac{\sqrt{3}}{2} \lambda\right)^{2} \\ &=\frac{m \lambda^{2}}{3}+\frac{m l^{2}}{3}+\frac{R\lambda^{2}}{12}+\frac{3 m l^{2}}{4}\end{aligned}$$
$$\begin{array}{l} =\frac{3}{2} m d^{2} \\\text { Hence option } C \text { is correct }\end{array} $$
Question 9
1 / -0

Two circular plates of radii 8cm and 6cm cut from a uniform metal sheet . They are placed side by side , touching each other lying in the same plane . The position of the centre of mass of the system from the centre of the larger disc is

A
4.08 cm

B
5.4 cm

C
5.04 cm

D
6.2 cm
Question 10
1 / -0

A small uniform sphere of diameter d is rolling without slipping with in another static hollow sphere of inner diameter 6d, the velocity of the center of mass of the inner sphere being $${ V }_{ 0 }$$ in magnitude. The angular velocity of the smaller sphere about an axis passing through its C.M. perpendicular to its direction of motion is:

A
$$\dfrac { 2{ V }_{ 0 } }{ 5d } $$

B
$$\dfrac { 2{ V }_{ 0 } }{ d } $$

C
$$\dfrac { 12{ V }_{ 0 } }{ 5d } $$

D
$$\dfrac { 14{ V }_{ 0 } }{ 5d } $$