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The angular momentum of a system of particles is conserved

when no external force acts upon the system

when no external torque acts upon the system

when no external impulse acts upon the system

when axis of rotation remains same

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Explanation

$τ = \frac{d L}{d t} . I f τ = 0, L = c o n s \tan t$

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One hollow and one solid cylinder of the same radius roll down on a smooth inclined plane. Then the foot of the inclined plane is reached by

solid cylinder earlier

hollow cylinder earlier

simultaneously both

the heavier earlier irrespective of being solid or hollow

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Explanation

Since friction is absent, rolling cannot take place and both bodies slide

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A circular platform is mounted a frictionless vertical axle. Its radius R = 2m and moment of inertia about the axle are 200 ${kgm}^{2}$ . It is initially at rest. A 50 kg man on the edge of the platform and begins to walk along the edge at the speed of 1 $m s^{- 1}$ relative to the ground. Time taken by the man to complete one revolution is

$πs$

$\frac{3 π}{2} s$

$2 πs$

$\frac{π}{2} s$

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Explanation

The angular speed of the platform is given by the linear speed of the man divided by the radius of the platform (Ï‰ = v/R = 1/2 rad/s). The time period for one revolution is T = 2Ï€/Ï‰ = 2Ï€/(1/2) = 4Ï€ seconds.

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The motor of an engine is rotating about its axis with an angular velocity of 100 rpm. It comes to rest is 15 s, after being switched off. Assuming constant angular deceleration. What are the numbers of revolutions made by it before coming to rest?

12.5

32.6

15.6

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Explanation

$0 = ω_{0} - α t ∴ α = \frac{ω_{0}}{t} = \frac{(100 \times 2 π) / 60}{15} = 0.7 r a d / s^{2}$

Now angle rotated before coming to rest $θ$ = $\frac{ω_{0}^{2}}{2 α}$

$o r θ = \frac{{(100 \times 2 π / 60)}^{2}}{2 \times 0.7} = 78.33 r a d o r n u m b e r o f r o t a t i o n n = \frac{θ}{2 π} = 12.5$

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The moment of inertia of a loop of radius R and mass M, about any tangent line in its plane will be

$\frac{3 M R^{2}}{2}$

$\frac{M R^{2}}{2}$

$M R^{2}$

$\frac{M R^{2}}{4}$

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Explanation

Use theorem of paralle axes.

$I = \frac{M R^{2}}{2} + M R^{2} = \frac{3}{2} M R^{2}$

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A hollow sphere of diameter 0.2 m and mass 2 kg is rolling on an inclined plane with velocity v = 0.5 m/ s. The kinetic energy of the sphere is

0.1 J

0.3 J

0.5 J

0.42 J

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Explanation

$K_{N} = \frac{1}{2} {mv}^{2} (\frac{K^{2}}{R^{2}} + 1) = \frac{1}{2} \times 2 \times (0 .5)^{2} \times (\frac{2}{3} + 1) = 0 .42 J$

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A man of the mass M stands at one end of a plank of length L which lies at rest on a frictionless surfacé. The man walks to the other end of the plank. If the mass of the plank is $\frac{M}{3}$ , the distance that the man moves relative to the ground is

$\frac{3 L}{4}$

$\frac{L}{4}$

$\frac{4 L}{5}$

$\frac{L}{3}$

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Explanation

Let plank moves x distance x opposite direction. Then, the displacement of man relative to the ground will be, (L- x).
Applying

$\begin{aligned} m_{R} x_{R} & = m_{L} x_{L} \end{aligned} or \begin{aligned} M (L - x) & = \frac{M}{3} x \end{aligned} Solving this equation we get, x = \frac{3 L}{4} ∴ Displacement of man relative to ground = L - \frac{3 L}{4} = \frac{L}{4}$

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Ratio of total kinetic energy and rotational kinetic energy in the motion of a disc is

1:1

2:7

1:2

3:1

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Explanation

$\frac{T o t a l k i n e t i c e n e r g y}{R o t a t i o n a l k i n e t i c e n e r g y} = \frac{\frac{1}{2} m v^{2} (1 + \frac{K^{2}}{R^{2}})}{\frac{1}{2} m v^{2} \frac{K^{2}}{R^{2}}} = \frac{1 + \frac{K^{2}}{R^{2}}}{\frac{K^{2}}{R^{2}}} = \frac{1 + \frac{1}{2}}{\frac{1}{2}} = \frac{3 / 2}{1 / 2} = 3$

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A bomb of mass 9 kg explodes into two pieces of mass 3 kg and 6 kg. The velocity of 3 kg mass is 16 m/s. The velocity of 6 kg mass is

4 m/s

8 m/s

16 m/s

32 m/s

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Explanation

$p_{3} = p_{6} o r 3 \times 16 = 6 \times v o r v = 8 m / s . A s p_{i} = 0$ or both the pieces should have equal and opposite momentum.

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A cylinder is rolling over a surface. Which points on it move rectilinearly?

All points on the curved surface of the cylinder.

All points on the flat surfaces of the cylinder.

All points on the axis of the cylinder.

None of the above

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Explanation

Points on the axis move rectilinearly.

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