System of Particles and Rotational Motion

A solid cylinder having radius R and length L is slipping on a rough horizontal plane. At time t=0 the cylinder has a translational velocity v0=49 m/s, perpendicular to its axis and a rotational velocity v0/4R about the centre. The time taken by the cylinder to start rolling is seconds. (coefficient of kinetic friction K=0.25 and g=9.8 m/s2)

1. 15
2. 5
3. 10
4. 7.5

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are and respectively.

1. 0.128 Nm, 100
2. 0.0128 Nm, 50
3. 0.128 Nm, 50
4. 0.0128 Nm, 100

The position of center of mass of three masses 2 kg, 3 kg and 15 kg placed with respect to mid point (p) of normal bisector, as shown in the figure is . [figure]

1. ( 34, 1.25 )
2. ( 34, 1.0 )
3. (0, 0)
4. (1.25, 0)

A solid sphere (A) of mass 5m and a spherical shell (B) of mass m, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of aA and aB, respectively. The ratio of aA and aB is .

1. 5 : 21
2. 6 : 10
3. 21 : 25
4. 1 : 5

Two identical bodies A and B of equal masses have initial velocities v1=4 i m/s and v2=4 j m/s respectively. The body A has acceleration a1=6 i+6 j m/s2 while the acceleration of the other body B is zero. The centre of mass of the two bodies moves in path.

1. circular
2. parabolic
3. straight line
4. elliptical

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle 1 and in the next 2 s it rotates through an angle 2. The ratio 2 1 is .

1. 6
2. 3
3. 4
4. 13

An object of uniform density rolls up the curved path with the initial velocity v0 as shown in the figure. If the maximum height attained by an object is 7v0210g (g = acceleration due to gravity), the object is a . [figure]

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

A solid sphere of mass M and radius R is divided into two unequal parts. The smaller part having mass M/8 is converted into a sphere of radius r and the larger part is converted into a circular disc of thickness t and radius 2R. If I1 is moment of inertia of a sphere having radius r about an axis through its centre and I2 is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia I2/I1= .

1. 35
2. 70
3. 140
4. 210

The position of an object having mass 0.1 kg as a function of time t is given as r= (10t2 i+5t3 j ) m. At t=1 s, which of the following statements are correct? Choose the correct answer from the options given below:

1. A, B and C only
2. B, C and D only
3. A, C and D only
4. A, B and D only

A particle is rotating in a circular path and at any instant its motion can be described as = 5t440- t33. The angular acceleration of the particle after 10 seconds is rad/s2.

1. 150
2. 120
3. 130
4. 170

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is m. (Take g=10 m/s2) [figure]

1. 3.33
2. 3.12
3. 2.22
4. 1.42

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure. If the mass of each disc is 600 gm and applied torque between two discs is 43 105 dyne.cm, the angular acceleration of the discs about the given axis AB is rad/s2. [figure]

1. 100
2. 22
3. 27
4. 11

When the position vector r = x i + y j + z k changes sign as - r, which one of the following vector will not flip under sign change?

1. Velocity
2. Linear momentum
3. Acceleration
4. Angular momentum

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm. When released from rest the heavier mass is observed to fall 81 cm in 9 s. The rotational inertia of the pulley is kg m2. (g = 9.8 m/s2)

1. 8.3 10-3
2. 4.75 10-3
3. 1.86 10-2
4. 9.5 10-3

A thin uniform rod (X) of mass M and length L is pivoted at a height ( L3 ) as shown in the figure. The rod is allowed to fall from a vertical position and lie horizontally on the table. The angular velocity of this rod when it hits the table top, is . (g = gravitational acceleration) [figure]

1. 32 gL
2. 3gL
3. 3 2 gL
4. 1 2 gL

Two small balls with masses m and 2m are attached to both ends of a rigid rod of length d and negligible mass. If angular momentum of this system is L about an axis (A) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about A is :

1. 43 Lmd2
2. 32 Lmd2
3. 2L5md2
4. 2Lmd2

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (R < L) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M):

1. 34MR2 + 16ML2
2. 38MR2 + 712ML2
3. 34MR2 + 712ML2
4. 38MR2 + 16ML2

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is kg m2.

1. 0.18
2. 0.72
3. 0.36
4. 0.63

A uniform bar of length 12 cm and mass 20 m lies on a smooth horizontal table. Two point masses m and 2m are moving in opposite directions with same speed of v and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency . The ratio of v and is : [figure]

1. 33
2. 2 88
3. 32
4. 66

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are false
2. Statement I is false but Statement II is true
3. Statement I is true but Statement II is false
4. Both Statement I and Statement II are true

A uniform rod of mass m and length l is suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is . (g acceleration due to gravity) [figure]

1. mg/3
2. mg/4
3. mg
4. mg/2

Two cars A and B each of mass 103 kg are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds 72 km/h and 36 km/h. The magnitude of angular momentum of car A with respect to car B is J s.

1. 3.6 105
2. 105
3. 3 105
4. 2 105

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 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

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 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 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