A force \(\vec{F}=\alpha\hat i+3\hat j+6\hat k\) is acting at a point \(\vec{r}=2\hat i-6\hat j-12\hat k.\) The value of \(\alpha\) for which angular momentum is conserved about the origin is:
1. \(-1\)
2. \(2\)
3. zero
4. \(1\)

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)6.66 kgm²s^-2

(2)8.58 kgm²s^-2

(3)10.86 kgm²s^-2

(4)2.86 kgm²s^-2

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 rev/ s² is

(1)25N

(2)50N

(3)78.5N

(4)157N

The ratio of the accelerations for a solid sphere (mass m and radius R) rolling down an incline of angle  θ 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 muscle string tied to point Q as shown in the figure. When the string is cut, the initial angular acceleration of the rod is -

(1)3g/2L

(2)g/L

(3)2g/L

(4)2g/3L

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along

1. a line perpendicular to the plane of rotation

2. the line making an angle of $45°$ to the plane of rotation

3. the radius

4. the tangent to the orbit

Two persons of mass 55 kg and 65 kg respectively, are at the opposite ends of a boat.The length of the boat is 3.0m and weighs 100 kg.The 55 kg man walks up to the 65 kg man and sits with him.If the boat is in still water the centre of mass of the system shifts by 

(1)3.0m

(2)2.3m

(3)zero

(4)0.75m

The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through:
 
1. B                   
2. C
3. D                   
4. A

Three masses are placed on the x-axis: \(300\) g at the origin, \(500\) g at \(x =40\) cm, and \(400\) g at \(x=70\) cm. The distance of the center of mass from the origin is:

The moment of inertia of a thin uniform rod of 

mass M and length L about an axis passing

through its mid-point and perpendicular to its

length is ${\mathrm{I}}_0$. Its moment of inertia about an 

axis passing through one of its ends and

perpendicular to its length is

(1) ${\mathrm{I}}_0+{\mathrm{ML}}^2/4$

(2) ${\mathrm{I}}_0+2{\mathrm{ML}}^2$

(3) ${\mathrm{I}}_0+{\mathrm{ML}}^2$

(4) ${\mathrm{I}}_0+{\mathrm{ML}}^2/2$