1. A square of side L/2 is removed from one corner of a square plate that has sides of length L. The center of mass of the remaining plate moves from C to C'. The displacement of the x-coordinate of the center of mass (from C to C') is
(a) 12 2 L (b) L 12 1 (c) L 6 1 (d) L 8 2
2. A square plate has a diagonal of length 21 cm. One of it’s corner is cut along the line joining the midpoint of neighbouring two sides. The distance of new centre of mass from the original centre of mass is
(a) 0.25 cm (b) 1.75 cm (c) 1.25 cm (d) none of these
3. From a circle of radius a, an isosceles right angled triangle with the hypotenuse as the diameter of the circle is removed. The distance of the centre of mass of the remaining part from the centre of the circle is
(a) 3(– 1)a (b) 6 ) 1 ( a (c) ) 1 ( 3 a (d) ) 1 ( 3 a
4. Two thin uniform rods of same length but made up of different materials are kept as shown. If the meeting point is the origin of co-ordinates, then centre of mass co-ordinates can be
(a) (L/2, L/2) (b) (2L/3, L/2) (c) (L/3, L/3) (d) (L/3, L/6)
5. A body of mass 2 g, moving along the positive x-axis in gravity free space with velocity 20 cm/s explodes at x = 1 m, t = 0 into two pieces of masses 2 / 3 g and 4 / 3 g. After 5 s, the lighter piece is at the point (3 m, 2 m, – 4 m ). The position of the heavier piece at this moment (in m) is
(a) (1.5, –1, –2) (b) (1.5, –2,–2) (c) (1.5, –1, –1) (d) None
6. Mass m1 hits & sticks with m2 while sliding horizontally with velocity v = 6 m/s along the common line of centres of the three equal masses (m1= m2= m3= 4 kg). Initially masses m2 and m3are stationary and the spring is unstretched. Find the minimum kinetic energy (in J) of m2in the process.
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7. Two smooth circular cylinders, each of mass 100 kg and radius 25cm, are connected at their centers by a massless inextensible string of length 60cm and rest upon a horizontal plane, supporting above them a third cylinder of mass 200 kg and radius 25cm (see figure). Find the tension (in N) in the string.
8. Two particles A and B are separated from each other by a distance l. At time t = 0, particle A starts moving with uniform acceleration a along line perpendicular to initial line joining A and B. At the same moment, particle B starts moving with acceleration of constant magnitude b such that particle B always points towards the instantaneous position of A. (b > a). Find the distance (in m) travelled by B till the moment B converges with A. (Take b = 3 m/s2, a = 1 m/s2and l = 8 m)
1. In the figure shown a hole of radius 2 cm is made in a semicircular disc of radius 6 at a distance 8 cm from the centre C of the disc. The distance of the centre of mass of this system from point C is
(a) 4 cm (b) 8 cm (c) 6 cm (d) 12 cm
2. Two blocks (from very far apart) are approaching towards each other with velocities as shown in figure. The coefficient of friction for both the blocks is = 0.2. Linear momentum of the system is
(a) conserved all the time (b) never conserved (c) is conserved upto 5 seconds (d) none of these
3. In the above problem, distance covered by centre of mass before coming to rest is
(a) 25 m (b) 37.5 m
(c) 42.5 m (d) 50 m
4. A particle moves from rest at A on the surface of a smooth circular cylinder of radius r as shown. At B it
leaves the cylinder. The equation relating and is r A
B (a) 3sin2cos (b) 2sin3cos
(c) 3sin2cos (d) 2sin3cos
5. A rigid body can be hinged about any point on the x-axis. When it is hinged such that the hinge is at x, the moment of inertia is given by I = 2x2 – 12x + 27. The x-coordinate of
centre of mass is
(a) 2 (b) 0 (c) 1 (d) 3
6. A spring mass system arrangement in vertical plane is as shown. Initially springs are in natural length and blocks A & B are resting on the ground. Determine the minimum mass (in kg) of block C so that block A leaves contact with the ground after releasing block C. (Take m = 10 kg) m 3m K m0 A B 3K C
7. A particle is moving along a vertical circle of radius 20 m with a constant speed of 31.4m/s. Straight line ABC is horizontal and passes through the centre of the circle. A shell is fired from point A at the instant when particle is at C. If distance AB is
A B m 20 u C
PH 20 3 m and shell collides with the particle at B, find
(i ) the smallest possible value of the angle (in degree) of projection, (ii) the corresponding velocity (in m/s) of projection .
( = 3.14 and g = 10 ms-2
)
8. A particle starts to move along a straight line from rest with an initial acceleration of 6 m/s2. The acceleration decreases linearly with time and becomes zero in 10 seconds. Then particle continues to move uniformly. Time taken by it to cover 400 m from rest is
n 50
s. Find the value of n.
1. A square plate of mass M and edge L is shown in figure. The moment of inertia of the plate about the axis in the plane of plate passing through one of its vertex making an angle 15° from horizontal is L 15° axis (a) 12 2 ML (b) 24 11ML2 (c) 12 7ML2 (d) none of these
2. The figure shows an isosceles triangular plate of mass M and side l. The angle at the apex is 90°. The apex lies at the origin and the base is parallel to x–axis. The moment of inertia of the plate about the z-axis is (a) 12 2 Ml (b) 24 2 Ml (c) 6 2 Ml (d) none of these
3. In the above problem, the moment of inertia of the plate about the x-axis is (a) 8 2 Ml (b) 32 2 Ml (c) 24 2 Ml (d) 6 2 Ml
4. From an uniform disc of mass M, radius R, a disc of radius R/2 is cut. The moment of inertia of the remaining disc about the centre of mass of the remaining disc is
(a) 32 13MR2 (b) 96 13MR2 (c) 82 37MR2 (d) 96 37MR2
5. The moment of inertia of a ring about an axis passing through its centre perpendicular to its plane is I. The moment of inertia about a chord at a distance of one sixth of radius in the plane of the ring is
(a) 2 I (b) 24 19I (c) 36 19I (d) 19 3I
6. An L shaped thin uniform rod of total length 2l is free to rotate in a vertical plane about a horizontal axis at P as shown in the figure. The bar is released from rest. Neglect air and contact friction. Determine the angular velocity (in rad/s) at the instant when it has rotated through 90° and reached the dotted position as shown. (Take )
3 1
m l
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7. A small steel ball A is suspended by an inextensible thread of length l = 1.5 m from O. Another identical ball is thrown vertically downwards such that its surface remain just in contact with thread during downward motion and collides elastically with the suspended ball. If the suspended ball just completes vertical circle after collision, calculate the velocity (in cm/s) of the falling ball just before collision. (g = 10 ms-2)
O B
A
8. A block of mass 2 kg is hanging with two identical springs as shown in figure. Find the acceleration (in m/s2) of the block just at the moment at which
right spring breaks. 2kg
1. If a circular concentric hole is made in a disk then about its axis
(a) Moment of inertia remains same (b) Moment of inertia increases (c) Radius of gyration increases (d) Radius of gyration decreases 2. A thin plate of mass M, length L, and width 2d is mounted
vertically on a frictionless axle along the z-axis, as shown. Initially the object is at rest. It is then tapped with a hammer to provide a torque , which produces an angular impulse H about the z-axis of magnitude H
dt. The angular speed of the plate about the z-axis after the tap will be(a) 2 2Md H (b) 2 2 Md H (c) 3 2 Md H (d) 4 2 Md H
3. Let I1, I2 and I3be the moment of inertia of a uniform square plate about axes AOC, xDx’ and yBy’ respectively as shown in the figure. The moments of inertia of the plate I1: I2: I3are in the ratio
(a) 7 1 : 7 1 : 1 (b) 7 12 : 7 12 : 1 (c) 12 7 : 12 7 : 1 (d) 1 : 7 : 7
4. A uniform rod AB of weight W is free to move in a vertical plane about a smooth hinge at A, and is sustained in equilibrium by a force P acting along a string BCP passing over a smooth peg C as shown. AC being vertical. If AC be equal to AB, then the force P is (a) cos W (b) W cos (c) sin W (d) W sin 5. A uniform rod of mass m and length l is connected to a
block of mass 2m through a massless string passing over a frictionless pulley as shown. The tension at the mid point of rod is m 2m (a) 3 4mg (b) 3 2mg (c) 3 mg (d) 6 mg
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6. A boy swims in a straight line to reach the other side of a river. His velocity is 1
ms
5 and the angle of swim with shore is 30º. Flow of river opposes his movement at 2ms1.
If width of river is 200 m, find the distance (in m) where he reaches the other bank from O.
30º 2 ms–1 200m 5ms–1 ' O O
7. Two masses each of mass m are attached at mid point B and end point C of a massless rod AC which is hinged at A. It is released from horizontal position as shown. Find the force (in N) at hinge A when rod becomes vertical. (Take m = 5kg and g = 10 m/s2)
8. A uniform rod AB of length L and mass m is suspended freely at A and hangs vertically at rest when a particle of same mass is fired horizontally with speed v to strike elastically the rod at its mid point. The particle is brought to rest after the impact. Find the impulsive reaction (in N-s) at A. (Take mv = 8)
1. The moment of inertia of the pulley arrangement as shown in the figure is 4 kgm2. The radii of bigger and smaller pulleys are 2m
and 1m respectively. The angular acceleration of the pulley arrangement is
(a) 2.1 rad/s2 (b) 4.2 rad/s2
(c) 1.2 rad/s2 (d) 0.6 rad/s2
2. Two particles start moving from the same point along the same straight line. The first moves with constant velocity v and the second with constant acceleration a. During the time that elapses before the second catches the first, the greatest distance between the particles is (a) a v2 (b) a v 2 2 (c) a v2 2 (d) a v 4 2
3. An external device supplies constant power to a rotating system. If is angular speed and is torque on body, then
(a)
1 (b) t (c) t3/2 (d) all are correct
4. Two solid uniform disks of equal mass each are mounted to rotate about an axis fixed through the center of the disk. Each disk is initially at rest. The radii of the disks are r1< r2. A force F is applied to each disk at its edge for the same amount of time. Assume that friction at pulley axis is negligible. The correct statement about the kinetic energy (K) and magnitude of angular momentum (L) of the disks is
(a) L1= L2and K1< K2 (b) L1< L2and K1= K2 (c) L1< L2and K1 > K2 (d) L1< L2and K1< K2
5. Determine the speed (in cm/s) with which block B rises in figure if the end of the string at A is pulled down with a speed of 2 m/s.
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balance on a slope of inclination = 37° with the help of a thread fastened to its jacket. The cylinder does not slip on the slope. The minimum required coefficient of friction to keep the cylinder in balance when the thread is held vertically is
100 n
. Find the value of n.
7. A rod of mass m, length l is held at rest on a smooth horizontal surface at an angle of 45° with the vertical. If it is released from the given position then determine immediate acceleration (in m/s2) of end of rod which is in contact of surface. (g = 10 m/s2)
8. In the given figure, all surface are frictionless and strings pulleys are massless. Find the accelerations (in m/s2) of block B and the wedge. m m A B Wedge D m m C 2m
1. A cylinder is pulled horizontally by a force F acting at a point below the centre of mass of the cylinder, as shown in figure. The surface is rough enough to prevent sliding. The free body diagram that best represents the frictional force is
(a) (b) (c) (d) cannot be interpreted
2. A spool is pulled horizontally by a constant force F below the centre of mass. The surface is rough enough to prevent sliding. The free body diagram that best represents the frictional force is
(a) (b) (c) (d) cannot be interpreted
3. A spool is pulled vertically by a constant force F (< Mg) as shown in figure. The surface is rough enough to prevent sliding. The free body diagram that best represents the frictional force is
(a) (b) (c) (d) cannot be interpreted
4. A particle of mass m moves with speed 2v to the right, while another particle of the same mass m moves with speed v toward the left, as shown in figure I. Their paths are separated by a distance b. At t = 0, when they are both at x = 0, they stick to a pole of length b and of negligible mass. For t > 0, consider the system as a rigid body of two masses m separated by distance b, as shown in figure II. The correct relation for the motion after t = 0 of the particles initially at y = b/2 is (a) 2 , 2vt y b x (b) b vt b y b vt b vt x 0.5 sin 3 , 0.5 cos 3 (c) b vt b y b vt b vt x 0.5 0.5 sin 3 , 0.5 cos 3 (d) b vt b y b vt b vt x 0.5 0.5 sin 6 , 0.5 cos 6
5. Uniform rod is hinged at one end in horizontal position as shown in the figure. The other end is connected to a block through a massless string m as shown. The pulley is smooth and massless. Masses of block and rod is same and is equal to m. Then acceleration of block just after release from this position is
(a) 13 6g (b) 4 g (c) 8 3g (d) None of these
PH 6. A square plate of mass M and length a is
suspended in equilibirum between two rough vertical walls spaced slightly more than a, by the help of two blocks as shown. If one of the string connecting the plate and the block is cut, then determine the immediate acceleration (in m/s2) of the plate (the coefficient of friction between plate and balls is 1/3).
M
M/2 M/2
7. Two trains A and B are moving on same track in opposite direction with velocity 25 m/s and 15m/s respectively. When separation between them becomes 225 m, drivers of both the trains applies brakes producing uniform retardation in train A while retardation of train B increases linearly with time at the rate of 0.3 m/s2. Find the minimum retardation (in cm/s2) of train A to avoid collision.
8. A conveyer belt of length l moves with a velocity v. A block of mass m is pushed against the motion of conveyer belt with velocity v0 from end B. Co-efficient of friction
between block and belt is . Find the value of v0 (in m/s)
so that the amount of heat liberated as a result of retardation of the block by conveyer belt is maximum. (Take = 0.4, g = 10 m/s2 and l = 2m) v0 v l B A
1. Four solid spheres P, Q , R and S are made to move on a horizontal surface which is rough. Sphere P is given a spin and released, sphere Q is given a linear velocity and released, sphere R is given a reverse spin and a forward linear velocity and released, sphere S is given a forward spin and a forward linear velocity and released as shown in figure. The direction of frictional force f for P, Q, R and S will be along
(a) left for all
(b) right, left, right, left respectively (c) right, left, right respectively
(d) right, left, left and either right or left respectively v v v P Q R S(v r)
2. A time varying force F = 2t is applied on a spool rolling without slipping as shown in figure. The angular momentum of the spool at time t about bottommost point is
(a) R t rt 2 (b) 2 2 ) ( t r r R (c) (R + r)t2 (d) data is insufficient
3. A horizontal force F = mg/3 is applied on the upper surface of a uniform cube of mass m and side a which is resting on a rough horizontal surface having S = 1/2. The distance between lines of action of mg and normal reaction N is
(a) 2 a (b) 3 a (c) 4 a (d) 6 a
4. A homogeneous cubical brick lies motionless on a rough inclined surface. The half of the brick which applies greater pressure on the plane is
(a) left half (b) right half
(c) both applies equal pressure
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5. A particle is moving in x-y plane with y =x and vx 4 2t
2 where symbols have their usual meanings. The displacement versus time graph of the particle would be
(a) s t (b) s t (c) s t (d) s t
6. Two men, each of mass 75 kg, stand on the rim of a horizontal large disc, diametrically opposite to each other. The disc has a mass 450 kg and is free to rotate about its axis. Each man simultaneously start along the rim clockwise with the same speed and reaches their original starting points on the disc. The angle turned by the disc with respect to the ground is
5 n
. Find the value of n.
7. A rigid body in shape of a triangle has vA= 5 m/s , vB= 10 m/s . Find the velocity (in cm/s) of point C. ( 52.24)
8. A square plate ABCD of mass m and side l is suspended by the help of two ideal strings P and Q as shown. Determine the acceleration (in m/s2) of the corner A of the square just at the moment the string Q is cut. (g = 10 m/s2)
A B
D C
1. A rod is hinged at its centre and is rotated by applying a constant torque starting from rest. The power developed by the external torque varies with time as
(a) (b) (c) (d)
2. A pulley is hinged at the centre and a massless thread is wrapped around it. The thread is pulled with a constant force F starting from rest. As the time increases,
(a) its angular velocity increases, but force on hinge remains constant (b) its angular velocity remains same, but force on hinge increases (c) its angular velocity increases and force on hinge increases (d) its angular velocity remains same and force on hinge is constant
3. A ladder of length L is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment, the speed of the end in contact with the horizontal floor is v and the ladder makes an angle = 30° with the horizontal. Then the speed of the ladder’s center must be (a) 3 2v (b) 2 v (c) v (d) none of these
4. In the previous question, if dv/dt = 0, then the angular acceleration of the ladder when = 45° is (a) 2 2 2 L v (b) 2 2 2L v (c) 2[v2/L2] (d) none of these
5. A plank of mass M is placed over smooth inclined plane and a sphere is also placed over the plank. Friction is sufficient between sphere and plank. If plank and sphere are released from rest, the frictional force on sphere is
(a) up the plane (b) down the plane (c) horizontal (d) zero 6. A ring of mass m and radius R has three particles attached to
the ring as shown in the figure. The centre of the ring has a speed v0. The kinetic energy of the system is (Slipping is absent) (a) 6 mv02 (b) 2 0 12 mv (c) 4 mv02 (d) 8 mv02
7. A uniform chain of length L has one of its end attached to the wall at point A, while
4 3L
of the length of the chain is lying on table as shown in figure. The minimum co-efficient of friction between table and chain so that chain remains in
37°
PH equilibrium is
4 n
. Find the value of n.
8. Two bodies of masses m and 4 m are attached with string as shown in the figure. The body of mass m hanging from a string of length l is executing oscillations of angular amplitude 0, while the other body is at rest. The
minimum coefficient of friction between the mass 4 m and the horizontal surface if 0= 60° is
2 n
. Find the value of n.
m 4m
0 l
9. Two blocks of mass 3 kg and 6 kg are connected by an ideal spring and are placed on a frictionless horizontal surface. The 3 kg block is imparted a velocity 2 m/s towards left. Find the velocity (in m/s) of 6 kg block when the speed of 3 kg block is minimum.
3kg 6kg
