Circular Motion

Subject : PHYSICS

Class-XII

1

A stone is thrown horizontally with the velocity 15m/s. Determine the tangential and normal accelerations of the stone in 1 second after it begins to move.

2

A particle moves in the x-y plane with the velocity v  a ˆi  b t ˆj . At the instant t = a 3 b the magnitude of tangential, normal and total acceleration are _____, _______, & _________.

3

Two strings of length l = 0.5 m each are connected to a block of mass m = 2 kg at one end and their ends are attached to the point A and B 0.5 m apart on a vertical pole which rotates with a constant angular velocity  = 7 rad/sec. Find



the ratio

T1 T2 of tension in the upper string (T1) and the lower string (T2).

[Use g = 9.8 m/s2] 4

A block of mass m placed on a smooth horizontal surface is attached to a spring and is held at rest by a force P as shown. Suddenly the force P changes its direction opposite to the previous one. How many times is the maximum extension l2 of the spring longer compared to its initial compression l1?

5

A ball of mass 1 kg is released from position A inside a wedge with a hemispherical cut of radius 0.5 m as shown in the figure. Find the force exerted by the vertical wall OM on wedge, when the ball is in position B. (neglect friction everywhere). Take (g = 10 m/s2)

6

A particle is confined to move along the +x axis under the action of a force F(x) that is derivable from the potential U(x) =ax3bx. (a) Find the expression for F(x) (b) When the total energy of the particle is zero, the particle can be trapped with in the interval x=0 to x=x1. For this case find the values of x1. (c) Determine the maximum kinetic energy that the trapped particle has in its motion. Express all answers in terms a and b. At what value of x will the kinetic energy be maximum ?

7

Two blocks of mass m1=10kg and m2=5kg connected to each other by a massless inextensible string of length 0.3m are placed along a diameter of a turn table. The coefficient of friction between the table and m1 is 0.5 while there is no friction between m2 and the table. The table is rotating with an angular velocity of 10rad/sec about a vertical axis passing through its centre. The masses are placed along the diameter of the table on either side of the centre O such that m1 is at a distance of 0.124m from O. The masses are observed to be at rest with respect to an observer on the turn table. (i) Calculate the frictional force on m1 (ii) What should be the minimum angular speed of the turn table so that the masses will slip from this position. (iii) How should the masses be placed with the string remaining taut, so that there is no frictional force acting on the mass m1. A stone is launched upward at 45° with speed v0. A bee follows the trajectory of the stone at a constant speed equal to the initial speed of the stone. (a) Find the radius of curvature at the top point of the trajectory. (b) What is the acceleration of the bee at the top point of the trajectory? For the stone, neglect the air resistance.

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9

The blocks are of mass 2 kg shown is in equilibrium. At t = 0 right spring in fig (i) and right string in fig (ii) breaks. Find the ratio of instantaneous acceleration of blocks?

figure (i)

figure (ii)

10

A uniform rod of mass m length L is sliding along its length on a horizontal table whose top is partly smooth & rest rough with friction coefficient . If the rod after moving through smooth part, enters the rough with velocity v0. (a) What will be the magnitude of the friction force when its x length (< L) lies in the rough part during sliding. (b) Determine the minimum velocity v0 with which it must enter so that it lies completely in rough region before coming to rest. (c) If the velocity is double the minimum velocity as calculated in part (a) then what distance does its front end A would have travelled in rough region before rod comes to rest.

11

Tangential acceleration of a particle moving in a circle of radius 1 m varies with time t as (initial velocity of particle is zero). Time after which total acceleration of particle makes and angle of 30° with radial acceleration is (A) 4 sec (B) 4/3 sec (C) 22/3 sec (D)

12

2 sec

A particle is moving along the circle x2 + y2 = a2 in anticlockwise direction. The x–y plane is a rough horizontal stationary surface. At the point (a cos, a sin), the unit vector in the direction of friction on the particle is: (A) cos  ˆi  sin  ˆj



(B)  cos  ˆi  sin  ˆj



(C) sin  ˆi  cos  ˆj

(D) cos  ˆi  sin  ˆj

13

When a conservative force does positive work on a body (A) the potential energy increases (B) the potential energy decreases (C) total energy increases (D) total energy decreases

14

The work done by the force F  x 2 ˆi  y 2 ˆj around the path shown in the figure is



(A)

2 3 a 3

(C) a3

15

16

(B) zero

(D)

4 3 a 3

F = 2x2 – 3x – 2. Choose correct option (A) x = – 1/2 is position of stable equilibrium (C) x = – 1/2 is position of unstable equilibrium

(B) x = 2 is position of stable equilibrium (D) x = 2 is position of neutral equilibrium

A bob attached to a string is held horizontal and released. The tension and vertical distance from point of suspension can be represented by.

(A)

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(B)

(C)

(D)

2

17

The potential energy in joules of a particle of mass 1 kg moving in a plane is given by U = 3x + 4y, the position coordinates of the point being x and y, measured in metres. If the particle is initially at rest at (6,4), then (A) its acceleration is of magnitude 5 m/s2 (B) its speed when it crosses the y-axis is 10 m/s (C) it crosses the y-axis (x = 0) at y = –4 (D) it moves in a straight line passing through the origin (0,0)

18

A particle of mass m is at rest in a train moving with constant velocity with respect to ground. Now the particle is accelerated by a constant force F0 acting along the direction of motion of train for time t0. A girl in the train and a boy on the ground measure the work done by this force. Which of the following are INCORRECT? (A) Both will measure the same work (B) Boy will measure higher value than the girl (C) Girl will measure higher value than the boy (D) Data are insufficient for the measurement of work done by the force F0

19

A block of mass m = 4 kg is kept on wedge kept on horizontal surface. Wedge is attached with a spring balance, other end of which is fixed rigidly with wall. Angle of inclined surface of wedge is  = 45°. Assume length of inclined surface to be sufficiently long. Maximum reading of spring balance will be (All surfaces are frictionless) (A) 2 2 kgwt (B) 4 kgwt. (C) 2 kgwt (D) In sufficient informance

20

A block of mass m is kept on wedge B which is kept on smooth horizontal surface. The block is imparted velocity is such a way that it reaches to point Q of the wedge. Then the work done by normal force acting on block by wedge in path PQ in ground frame will be : (A) Positive (B) Negative (C) Zero (D) Positive in first half part there after negative

21

A block of mass ‘m’ is slowly moved from point A to point C. Applied force is always tangent to surface in contact. Coefficient of friction between block and surface is . Work done by force F in this process while moving block from point A to point C is : (A) ( mgl + mgh) (B) mgl + 2mgh (C) zero (D) mgh + 2 mgl

22

23

1 x   ˆi – 2 ˆj  N acts on it. Work done by A particle moves from point (4m, 2m) to (1m 1m) while a force  = F y y  force is equal to : (A) 1 J (B) –1J (C) 2 J (D) Cann’t be calculated without knowledge of path followed b/w the two points.

A block of mass m is placed on plank as shown in figure. There is no friction at any contact. Then, find minimum value of acceleration a0 so that block will just reach topmost point P(a0 is constant acceleration of plank). (A) 2g

24

(C) g

(D)

2g(2  R) R

A block of mass, attached with three spring constant K, 2K and K, is kept on inclined plane of angle of inclination 30° with the horizontal. Initially spring is in natural length. Block is released from rest. The distance traveled by block till it reaches to initial position for the first time will be : (A)

25

(B) g/2

mg 8K

(B)

mg 4K

(C)

mg 2K

(D)

A block B of mass m is pulled with a force F. The coefficient of friction between block A and the curved surface is  =

1

. Initially spring is in natural length. Then, the minimum value of F so that block A will just start 3 moving will be (mass of block A = 2kg) : (A) 20 N (B) 10 N (C) 40 N (D) 10/3 N

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26

A block of mass m is kept at rest at the top of a smooth hemisphere as shown in figure. ‘ O’ is centre of hemisphere. Line OB makes an angle ‘’ with vertical. The hemisphere is accelerated towards right with acceleration ‘a’. Velocity of hemisphere is ‘V0’ at the time block reaches to the position B. Work done by hemisphere on the block as it reaches to position B is : 1 2 (A) Greater than maR sin   mv 0 2

(C) mgR(1 – cos ) + maR sin  –

(B) {mgR (1 – cos q) + maR sin }

1 mV02 2

(D) Less than maR sin  +

27

A block of mass m1 = 5 kg is tied with a light string which passes over an ideal pulley. A sleeve of mass m2 = 3kg which can slip on a smooth vertical rod is tied at the other end of string. Initially sleeve is kept at position ‘A’ where string makes an angle 37° with the rod. Sleeve is released from this position. Tick the correct statement for subsequent motion. (A) Speed of sleeve will increase and becomes maximum when string between pulley & rod becomes horizontal. (B) Part of string between pulley & rod can become horizontal for any value of m1 & m2 (C) Speed of sleeve becomes maximum at a distance 0.7 m from A (D) Speed of sleeve becomes maximum at a distance 0.9 m from A.

28

A bead of mass 2kg can freely move along thin rigid wire ABC. Wire lies in vertical plane. Bead is released from point A. Bead takes 3 seconds and 8 seconds to reach point B and point C respectively. Then, the average power delivered by resultant of normal force and weight acting on bead between B and C will be : (A)

29

25 w 2

100 w 3

(D) 28 w

(B) 400 watts

(C)

200 5

watts

(D)

900 5

watts

3t 2 watt, where t is in second. If velocity of 2 particle at t = 0 is V = 0. The velocity of particle at t = 2 sec. will be :

The power supplied to a particle of mass 2 kg varies with time as P 

(A) 4 m/s 31

(C)

A balloon is moving horizontally at a constant speed of 10 m/sec.. A particle of mass 2kg is released from it. Then, instantaneous power delivered by weight of particle after time 2 second from the instant of release will be : (A) 200 watts

30

(B) 16 w

(B)

2 m/s 3

(C) 2 m/s

(D)

4 m/s 3

A particle of mass m is moving in a circular path of constant radius (r = 1m) such that its centripetal acceleration ac is varying with time as ac = K2rt2, where k is constant. Then power delivered to the particle by the force acting on it at t = 5 second would be (take mk2 = 1 unit) (A)

5 watt 2

(B) 5 watt

(C) 4 watt

(D) 3 watt

32

A block of mass 2 kg is pulled up on a smooth incline of angle 30° with the horizontal. If the block moves with an acceleration of 1 m/s2, then the power delivered by the pulling force at a time t = 4 sec. after the motion starts is : (A) 12 watt (B) 48 watt (C) 24 watt (D) 24/3 watt

33

Which of the following represents conservative force. 

(A) F  k(xiˆ  xyj) ˆ



(B) F  –k(yiˆ  xj) ˆ



(C) F  k(xyiˆ  xyj) ˆ



(D) F  –k(y2 ˆi  xj) ˆ

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Answer Key 2g 3g 1. at= , an= 13 13 6. F = 3ax2 + b, x =

8. (a)

V02

2g 11. (C) 18. (A, C) 25. (B) 32. (B)

, (b) 2g

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12. (C) 19. (B) 26. (D) 33. (B)

2.

3b 2 , b 2 , b

2b b , KEmax = a 3 3

b b , x a 3a

9. 25/24 13. (B) 20. (B) 27. (C)

3. 9

5.

15 3 N 2

7. (i) 36N, (ii) 11.66rad/sec ,(iii) 0.1m, 0.2m

10. (a) f = 14. (B) 21. (D) 28. (B)

4. 3

15. (A) 22. (B) 29. (B)

m xg ; (b)  16 (A) 23. (A) 30. (C)

5 2 17. (A, B, C) 24. (C) 31. (B)

g ; (c)

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