System of Particles and Rotational Motion

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M. Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitude of the acceleration experienced by the blocks is (assume no slipping of string on pulley). [figure]

1. (M - m)g [ ( 83 )M + m ]
2. (M - m)g2M + m
3. (M - m)gM + m
4. (M - m)g [ ( 136 )M + m ]

A rod of linear mass density ' ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameters is:

1. L38 2
2. L316 2
3. L34 2
4. L312

A thin solid disk of 1 kg is rotating along its diameter axis at the speed of 1800 rpm. By applying an external torque of 25 N m for 40 s, the speed increases to 2100 rpm. The diameter of the disk is m.

A rod of length 5 L is bent at right angle keeping one side length as 2 L. [figure] The position of the centre of mass of the system : (Consider L = 10 cm)

1. 4 i + 9 j
2. 2 i + 3 j
3. 5 i + 8 j
4. 3 i + 7 j

A, B and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure. [figure] The moment of inertia of the given system about PQ axis is x15I, where I is the moment of inertia of the disc about its diameter. The value of x is .

Let M and R be the mass and radius of a disc. A small disc of radius R/3 is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis passing through the centre O and perpendicular to the plane of disc is 4xMR2. The value of x is . [figure]

Which of the following are correct expressions for torque acting on a body? ( r = position vector; p = linear momentum; L = angular momentum; = angular acceleration; I = moment of inertia; F = force; t = time) Choose the correct answer from the options given below:

1. A, B, D and E Only
2. C and D Only
3. B, C, D and E Only
4. B, D and E Only

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is x5 where x = .

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is 8 m/s. The speed of the particle on the rim of the wheel at the same level as the centre of the wheel, will be :

1. 4 2 m/s
2. 8 m/s
3. 4 m/s
4. 8 2 m/s

A solid sphere with uniform density and radius R is rotating initially with constant angular velocity ( 1) about its diameter. After some time during the rotation it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes R/2 is x 1. The value of x is .

A force of 49 N acts tangentially at the highest point of a sphere (solid) of mass 20 kg, kept on a rough horizontal plane. If the sphere rolls without slipping, then the acceleration of the center of the sphere is [figure]

1. 2.5 m/s2
2. 3.5 m/s2
3. 0.25 m/s2
4. 0.35 m/s2

Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is ' '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

1. /4
2. /8
4. /2

A square lamina OABC of length 10 cm is pivoted at 'O'. Forces act at lamina as shown in figure. If lamina remains stationary, then the magnitude of F is : [figure]

1. 10 N
2. 0 (zero)
3. 10 2 N
4. 20 N

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m, would be: [figure]

1. 20 rad/s
2. 30 rad/s
3. 10 rad/s
4. 0 rad/s

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is :

1. 38Mr2
2. 2Mr2
3. 12Mr2
4. 32Mr2

[figure] A wheel of radius 0.2 m rotates freely about its center when a string that is wrapped over its rim is pulled by a force of 10 N as shown in figure. The established torque produces an angular acceleration of 2 rad/s2. Moment of inertia of the wheel is kg m2. (Acceleration due to gravity = 10 m/s2)

The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of F = i - j + k acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is .

[figure] Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t=0, they are given an initial velocity VA=V0 AC, VB=V0 BA and VC=V0 CB. Here, AC, CB and BA are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is :

1. 12 a mV0
2. 3 a mV0
3. 32 a mV0
4. 32 a mV0

The center of mass of a thin rectangular plate (fig - x) with sides of length a and b, whose mass per unit area ( ) varies as = 0 xab (where 0 is a constant), would be . [figure]

1. ( 23a, 23b )
2. ( 23a, b2 )
3. ( 13a, b2 )
4. ( a2, b2 )

Two iron solid discs of negligible thickness have radii R1 and R2 and moment of inertia I1 and I2, respectively. For R2 = 2R1, the ratio of I1 and I2 would be 1/x, where x = .

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is n times higher than the moment of inertia of the given ring. Here, n = . Consider all the bodies have equal masses.

A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

1. 290 g
2. 200 g
3. 190 g
4. 300 g

An object of mass 'm' is projected from origin in a vertical xy plane at an angle 45 with the x-axis with an initial velocity v0. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

1. mv032 2 g along negative z-axis
2. mv032 2 g along positive z-axis
3. mv034 2 g along positive z-axis
4. mv034 2 g along negative z-axis

A uniform solid cylinder of mass 'm' and radius 'r' rolls along an inclined rough plane of inclination 45 . If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

1. 2 g
2. 1 2 g
3. 13 2 g
4. 2 g3

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be t1 and t2, respectively, then

1. t1 < t2
2. t1 = 2t2
3. t1 > t2
4. t1 = t2

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

1. 34
2. 43
3. 52
4. 25

A solid sphere of mass ' m ' and radius ' r ' is allowed to roll without slipping from the highest point of an inclined plane of length ' L ' and makes an angle 30 with the horizontal. The speed of the particle at the bottom of the plane is v1. If the angle of inclination is increased to 45 while keeping L constant. Then the new speed of the sphere at the bottom of the plane is v2. The ratio v12:v22 is

1. 1:3
2. 1:2
3. 1: 2
4. 1: 3

Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be

1. 1.5 cm
2. 2.0 cm
3. 0.5 cm
4. 1.0 cm

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that (t)=5t2-8t, where (t) is the angular position of the rotating disc as a function of time t. How much power is delivered by the applied torque, when t=2 s ?

1. 60 MR2
2. 72 MR2
3. 8 MR2
4. 108 MR2

A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above. [figure]

1. 1732MR2
2. 1332MR2
3. 932MR2
4. 732MR2