Answer:
1. Answer: (a) 1 m/s
Explanation: The velocity is given as,
\(\frac{1}{2}\omega^2=\frac{1}{2}mv^2\)
\(\frac{1}{2}\times3(2)^2=\frac{1}{2}\times12\times v^2\)
v = 1 m/s
2. Answer: (c) g/4
Explanation: \(a=\frac{gsin\theta}{1+\frac{k^2}{R^2}}\)
= g sin 30° / 1+ 1
= \(\frac{g}{2}\times\frac{1}{2}\)
= g/4
3. Answer: (d) Axis of rotation
Explanation: When a mass is rotating in a plane about a fixed point, its angular momentum is directed along the axis of rotation.
4. Answer: (d) All of these
Explanation: Moment of inertia of a body depends on the mass of the body, its shape and size, distribution of mass about the axis of rotation, and the position and orientation of the axis of rotation.
5. Answer: (a) Kg
Explanation: The SI Unit of mass is Kilogram(Kg) and the CGS unit is gram(g).
6. Answer: (b) Angular momentum remains constant
Explanation: Angular momentum will remain the same since external torque is zero.
7. Answer: (d) Both b and c
Explanation: Particle moves in a spiral path with decreasing radius. The directon of angular momentum remains constant.
8.Answer: (a) t = Ia
Explanation: Torque = Product of MOI \(\times\) Ratio of change of ω
∴ t = Ia
9. Answer: (c) may lie within, outside on the surface of the body
Explanation: The position of centre of mass depend upon (1) its shape and (2) the way mass distributed on its shape. These two factor decide whether centre of mass of gravity lie inside the body or outside the body.If the solid body has a regular structure and its mass is distributed uniformly over its body (i.e. for symmetrical objects) then its centre of gravity must lie inside the body.But if solid body have irregular structure and mass is not distributed uniformly then centre of mass may lie inside the body or may outside the body.
10. Answer: (a) relative distance between the particles
Explanation: The resultant of all forces, on any system of particles, is zero. Therefore their centre of mass does not depend upon the forces acting on the particles.
11. Answer: (a) total external forces
Explanation: The motion of centre of mass depends only on the net external force.
Fexternal = MaCM
12. Answer: (a) joining the particles
Explanation: Centre of mass lies on the line joining the centre of masses.
13. Answer: (d) Both b and c
Explanation: In the fixed axis rotation we see that every point on the body has two components of velocity, one in the radial direction and on in the tangential direction. The resultant of these velocities is not same for any two points lying in the plane of the body. Any two points on the radial line have the radial acceleration directed towards the center of equal magnitude and the the tangential acceleration of equal magnitude as well. Thus option B is correct. All the particles lying on the curved surface of a cylinder whose axis coincides with the axis of rotation have the same speed but different velocities.
14. Answer: (a) in all cases
Explanation: The motion of centre of mass of the body depends only on external forces.Thus COM of the stick will move along a parabolic path in all cases .
15. Answer: (a) rotational motion
Explanation: A couple mainly comprises of two parallel forces which are both equal and opposite. There is no line of action and hence it produces only rotational motion.
16. Answer: (a) It is directly proportional to moment of inertia
Explanation: From L = Iω, we find that angular momentum is directly proportional to the moment of inertia. Moreover, angular momentum is a vector quantity.
17. Answer: (a) Transverse component
Explanation: The radial component of the force has no contribution to the torque because it passes through the pivot point. So, it is only the tangential component of the force which affects torque (since it is perpendicular to the line between the point of action of the force and the pivot point).
18. Answer: (a) turning moment of force
Explanation: Turning moment of a force= force x distance(r) from the axis of rotation. To produce a given turning moment, force required is smaller, when r is large. That’s what happens when handle of a screw is made wide.
19. Answer: (b) Decreases initially and increases again
Explanation: According to Law of conservation of angular momentum gives
Iω = constant
When viscous liquid dropped at the center of horizontal table is made to spread out, then its moment of inertia increases and so, from Equation its angular velocity decreases. But when it starts falling then its moment of inertia starts decreasing and so, its angular velocity increases.
20. Answer: (c) his moment of inertia decreases
Explanation: According to conservation of Angular Momentum, Iω = constant
Hence, when a person sitting on a rotating stool suddenly lowers his hands, then his angular velocity will increase and moment of inertia will decrease.
21. Answer: (c) his moment of inertia decreases
Explanation: In a whirlwind in a tornado, the air from nearby regions gets concentrated in a small space thereby decreasing the value of its moment of inertia considerably. Since, Iω = constant, so due to decrease in moment of inertia of the air, its angular speed increases to a high value.
22. Answer: (c) If the assertion is true but the reason is false
Explanation: In orbital motion, the angular momentum vector is perpendicular to the plane in which the rotational motion take place and its sense is given by the right hand fist rule. When the fingers of right hand fist point in the direction of motion, the thumb is in the direction of L (angular momentum).
23. Answer: (c) angular velocity
Explanation: Moment of inertia of a body does not depend on the angular velocity of the body.
24. Answer: (a) the angular velocity increases
Explanation: As the gymnast lowers his hands, the radius of gyration decreases and hence moment of inertia also decreases. Angular momentum remains conserved and hence angular velocity increases.
25. Answer: (c) the total momentum of the ball and the earth is conserved
Explanation: Momentum of the system changes only due to external force (OR impulse).
When a ball hits the floor and gets rebound, then no external force (Impulse) acts on the ball as well as the floor, thus total momentum of the "ball + earth" system remains conserved. Also some of the mechanical energy of the ball is always lost in an inelastic collision due to its deformation, hence it cannot remain conserved in this collision.
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