System of Particles and Rotational Motion · Question 36

Point masses m1 and m2 are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity 0 is minimum, is given by [figure]

1. x = m2m1L
2. x = m2Lm1 + m2
3. x = m1Lm1 + m2
4. x = m1m2L

An automobile moves on a road with a speed of 54 km h-1. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m2. If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is

1. 10.86 kg m2 s-2
2. 2.86 kg m2 s-2
3. 6.66 kg m2 s-2
4. 8.58 kg m2 s-2

A force F = i + 3 j + 6 k is acting at a point r = 2 i - 6 j - 12 k. The value of for which angular momentum about origin is conserved is

1. zero
2. 1
3. -1
4. 2

A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is

1. W(d-x)x
2. W(d-x)d
3. Wxd
4. Wdx

A mass m moves in a circle on a smooth horizontal plane with velocity v0 at a radius R0. The mass is attached to a string which passes through a smooth hole in the plane as shown. The tension in the string is increased gradually and finally m moves in a circle of radius R02. [figure] The final value of the kinetic energy is

1. 2mv02
2. 12mv02
3. mv02
4. 14mv02

Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is [figure]

1. 165mr2
2. 4mr2
3. 115mr2
4. 3mr2

A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound round the cylinder with one end attached to it and other hanging freely. Tension in the string required to produce an angular acceleration of 2 revolutions s-2 is

1. 25 N
2. 50 N
3. 78.5 N
4. 157 N

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle without slipping and slipping down the incline without rolling is

1. 5 : 7
2. 2 : 3
3. 2 : 5
4. 7 : 5

A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in figure. When string is cut, the initial angular acceleration of the rod is [figure]

1. 2gL
2. 2g3L
3. 3g2L
4. gL

A small object of uniform density rolls up a curved surface with an initial velocity 'v'. It reaches upto a maximum height of 3v24g with respect to the initial position. The object is

1. hollow sphere
2. disc
3. ring
4. solid sphere