System of Particles and Rotational Motion Test

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System of Particles and Rotational Motion Test - 61

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Question 1
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Three-point masses $$ m_1, m_2\ and\ m_3 $$ are located at the vertices of an equilateral triangle of side $$ á'$$ what is the moment of inertia of the system about an axis along the altitude of the triangle passing through $$ m_1 $$?

A
$$ (m_1 +m_2) \dfrac {a^2}{4} $$

B
$$ ( m_2 +m_2) \dfrac {a^2}{4} $$

C
$$ ( m_1+ m_3) \dfrac {a^2}{4} $$

D
$$ (m_1+ m_2 + m_3) \dfrac {a^2}{4} $$

Solution

Since the axis passes through $$m_1$$ and it's a point mass, it 's contribution to moment of Inertia is zero .
For $$m_1$$ and $$m_2$$ distance from the axis is $$\dfrac{a}{2}$$ as altitude divides the base of equilateral triangle into half.
So, $$I=(m_2+m_3)\dfrac{a^2}{4}$$
Hence Option B is correct
Question 2
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Two identical rods are joined to form an $$'X'$$. The smaller angle between the rods is $$\theta$$. The moment of inertia of the system about an axis passing through point of intersection of the rods and perpendicular to their plane is proportional to :

A
$$\theta$$

B
$$\sin^2 \theta$$

C
$$\cos^2 \theta$$

D
independent of $$\theta$$

Solution

The moment of inertia of the two rods will always be the summation of their individual moment of inertia irrespective of the angle between them.
$$I_{total}=I_A+I_B$$

$$I_{total}=\dfrac{ml^2}{12}+\dfrac{ml^2}{12}$$

Hence correct option is D, independent of $$\theta$$
Question 3
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Directions For Questions

A string wrapped around a cylinder of mass $$m$$ and radius $$R$$. The end of the string is connected to block of same mass hanging vertically. No friction exists between the horizontal surface and cylinder.

...view full instructions

Distance moved by cylinder during time taken by it to complete one rotation is:

A
$$2\pi R$$

B
$$\pi R$$

C
$$3\pi R$$

D
$$\dfrac{4\pi R}{3}$$

Solution

Total distance moved by the string:
$$s=ut+\dfrac{1}{2}at^2$$

$$s=\dfrac{1}{2}\times \dfrac{3g}{4}\times \dfrac{8\pi R}{g}=3\pi R$$

Distance moved in unwinding:
$$s'=\dfrac{1}{2}\times \dfrac{g}{4}\times \dfrac{8\pi R}{g}=\pi R$$

Distance moves by the cylinder in one complete rotation= $$3\pi R-\pi R=2\pi R$$
Question 4
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A uniform rod of length $$l$$ and mass $$m$$ makes constant angle $$\theta$$ with an axis of rotation, which passes through one end of the rod. Its moment of inertia about this axis is :

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

B
$$\dfrac{ml^2}{3}\sin \theta$$

C
$$\dfrac{ml^2}{3}\sin^2\theta$$

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

Solution

consider a infinitesimally small element of length $$dx$$ having mass $$dm=\dfrac{m}{l}dx$$ and at distance $$x$$
moment of inertia of this small element $$dI=dmr^2$$
where $$r=x\sin\theta$$ and $$dm=\dfrac{m}{l}dx$$
$$dI=\dfrac{m}{l}\sin\theta x dx $$
to find total moment of inertia, integrate wrt x:

$$\int dI=\int_{0}^{l}\dfrac{m}{l}\sin\theta x dx $$

$$I=\dfrac{m}{l}\sin\theta \int_{0}^{l}x dx $$

$$I=\dfrac{1}{3}ml^2\sin\theta$$
Question 5
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Two particles A and B, initially at rest, move 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 of the system is

A
3V

B
V

C
1.5 V

D
Zero

Solution

We know that $$ F_{ext} = Ma_{cm} $$
We consider the two particles in a system . mutual force of attraction in an internal force . there are no external forces acting on the system from Eq(I)

we get
$$ A_{CM} = 0$$

Since intial $$ v_{CM} = 0 $$

therefore final $$v_{CM} = 0 $$
Question 6
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Directions For Questions

A string wrapped around a cylinder of mass $$m$$ and radius $$R$$. The end of the string is connected to block of same mass hanging vertically. No friction exists between the horizontal surface and cylinder.

...view full instructions

Acceleration of cylinder is :

A
$$g$$

B
$$\dfrac{g}{2}$$

C
$$\dfrac{g}{4}$$

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

Solution
Question 7
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Directions For Questions

A string wrapped around a cylinder of mass $$m$$ and radius $$R$$. The end of the string is connected to block of same mass hanging vertically. No friction exists between the horizontal surface and cylinder.

...view full instructions

Distance moved by hanging mass during the above time interval is :

A
$$2\pi R$$

B
$$\pi R$$

C
$$3\pi R$$

D
$$\dfrac{4\pi R}{3}$$

Solution

$$s=ut+\dfrac{1}{2}at^2$$

$$s=\dfrac{1}{2}\times \dfrac{3g}{4}\times \dfrac{8\pi R}{g}=3\pi R$$
Question 8
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Directions For Questions

A long slender rod of mass 2 kg and length 4 m is placed on a smooth horizontal table. two particles of masses 2 kg and 1 kg strike the rod simultaneously and stick to the rod after collision as shwon in fig.

...view full instructions

Velocity of the center of mass of the rod after collision is

A
12 m/s

B
9 m/s

C
6 m/s

D
3 m/s

Solution
Question 9
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A wire of length $$l$$ and mass $$m$$ is first bent into a circle, then in a square and then in an equilateral triangle. The moment of inertia in these three cases about an axis perpendicular to their planes and passing through their centers of masses are $$I_1$$ $$I_{2}$$ & $$I_{3}$$ respectively. Then maximum of them is:

A
$$I_{1}$$

B
$$I_{2}$$

C
$$I_{3}$$

D
data insufficient

Solution
Question 10
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For which of the following does the centre of mass lie outside the body?

A
A pencil

B
A shot put

C
A dice

D
A bangle

Solution

A bangle is a ring like shape and the centre of mass of ring lies at its centre which is outside the ring or bangle. Hence verifies the option (d).