Laws of Motion

Study Material

Laws of Motion & Friction

Nothing in JEE will beyond this model questions in Newton’s Laws of motion & friction Note: →

1)

→

Spring force: F = − k x x is displacement of the free end from its natural length or deformation of the spring where K=spring constant.

2)

Spring Property: K × l = constant = Natural length of spring.

3)

If spring is cut into two ratio m:n then spring constant is given by

l1 =

ml ; m+n

l2 =

nl m+n

k l = k1l 1 = k 2 l 2

For series combination of springs

1 1 1 = + + ........ k eq k1 k 2

For parallel combination of spring

k eq k1 + k 2 + k 3 …….

4)

Spring Balance: It does not measure the weight. It measures the force exerted by the object at the hook.

5)

String Constraint: When two objects are connected through a string and if the string have the following properties. a) The length of the string remains constant i.e. inextensible string. b) Always remains tight, does not slacks. Then the parameters of the motion of the objects along the length of the string and in the direction of extension have definite relation between them.

6)

Wedge Constraint: Conditions: i) There is a regular contact between two objects ii) Objects are rigid. The relative velocity perpendicular to the contact plane of the two rigid objects is always zero if there is a regular contact between the objects. Wedge constraint is applied for each contact.

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In other words, Components of velocity along perpendicular direction to the contact plane of the two objects is always equal if there is no deformations and they remain in contact. 7)

Newton’s Law for a System: →

→

→

→

F ext = m1 a 1 + m2 a 2 + m3 a 3 + .......... →

F ext = Net external force on the system. m1 , m2 , m3 are the masses of the objects of the system and →

→

→

a 1 , a 2 , a 3 are the acceleration of the objects respectively

8)

Newton’s Law for non inertial frame: →

→

→

F Re al + F Pseudo = m a

Net sum of real and pseudo force is taken in the resultant force. →

a = Acceleration of the particle in the non inertial frame →

→

F Pseudo = − m a Frame Pseudo force is always directed opposite to the direction of the acceleration of the frame. Pseudo force is an imaginary force and there is no action-reaction for it. So it has nothing to do with Newton’s Third Law Reference Frame: A frame of reference is basically a coordinate system in which motion of object is analyzed. There are two types of reference frames. a) Inertial reference frame: Frame of reference moving with constant velocity. b) Non-inertial reference frame: A frame of reference moving with non-zero acceleration.

Study Material 9)

Tension in a Spring: Attached between masses remains unchanged during infinitely small time interval because it requires the displacement of the masses to change the tension and which requires a finite time interval. When one end of the spring is made free then the tension becomes zero in infinitely small interval.

Multiple choice questions (only one is correct) 1.

A body of specific gravity 6, weighs 0.9kg when placed in one pan (say pan A) and 1.6kg when placed on the other pan (pan B) of a false balance. The beam is horizontal when both the pans are empty. Now if the body is suspended from pan A and fully immersed in water, it will weigh a) 0.5kg

2.

b) 0.6kg

c) 0.75kg

d) 0.8kg

The pulley has mass M >m. String is massless. The above system is released from rest from the position shown. Then

a) Body (1) will slowly come down, till equilibrium is attained at level AA’ shown. b) The system will perform oscillations with equilibrium position at level AA’ and amplitude 0.5m c) The response of the system depends on whether pulley-string interface has friction or not. d) The system will continue to be in the same initial position.

3.

In the position shown above Let T 1 be the tension in the left part of the string, T2 = Tension in the right part, N = Normal reaction on block (2) by the resting surface. Then

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a) T1 = T2 = 0 and N = mg b) T1 =mg, T2 = 0, N =mg c) T 1 = T2 = mg and N = 0 d) T1 = T2 =

mg mg and N= 2 2 →

4.

A body of mass m was slowly hauled up the hill and down the hill onto the other side by a force by a F which at each point was directed along a tangent to the trajectory. The work performed by this force, if the coefficient of friction is µ1 uphill and µ

5.

2

downhill, is

a) mg (µ1l 1 + µ 2 l 2 )

b) mg (2h + µ1l 1 + µ 2 l 2 )

c) mg (h + µ1l 1 + µ 2 l 2 )

d) mg (µ1l 1 − µ 2 l 2 )

The bob of a pendulum is taken to position A and given an initial velocity u in the direction shown, in the following cases.

Study Material If the minimum value of u so that the pendulum reaches position OB in case (i) and position OC in case (ii) are u1 in case (i) & u2 in case (ii), then

a)

6.

(3 + 2 )

b)

(

u2 is g = 10 ms − 2 u1

(2 + 3 )

)

c)

(2 − 3 )

d)

(3

2−2

)

A conveyor belt carrying powdery material is at an angle 370 to the horizontal and moves at a constant speed of 1ms-1 as shown. Through a small hole in the belt, the powdery material drops down at a constant rate of 1kg per second. What is the force to be applied on the belt along the direction of its motion so as to maintain its constant speed of 1ms-1?

a) -1 N

7.

b) Zero

c) +1 N

d) None of the above

Small body A on a hemispherical body B which is on a wedge C which is on a smooth horizontal surface. System is released from rest from the position shown when α = 37 0 , l B = 10cm, l = 5cm . When α = 530 , l C is 4.5cm l B at that instant is (neglect friction)

a) 9 cm

8.

b) 10 cm

c) 10.5 cm

d) 11 cm

A pendulum consists of a mass m attached to the end of a light string 0.5m long. It can oscillate in the vertical plane. If it is let go in the horizontal position and has an inelastic collision with the floor, e = 0.5, the rebound velocity is (ms-1) (the angle turned is π / 6 ) (g = 10ms-2)

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a)

9.

5 2

b)

5 4

35 4

c)

45 4

d)

The track is in the vertical plane. The track is rough with friction coefficient µ . A particle at A is allowed to slide down and goes upto B and returns. Heights of A and B are 0.2m and 0.1m respectively, as shown. The maximum possible value of µ is

a)

10.

1 4

b)

1 7

c)

1 6

d)

1 5

A particle is projected from a point on a horizontal floor. After it has three collisions with the floor, it is found

 1012   . The coefficient of restitution is 36  2 

that the ratio of maximum height to minimum peaks reached by it is  a) .5

11.

b) 0.64

c) 0.8

d) 0.9

The potential energy function along the positive x axis is given by U ( x ) = − ax +

b , a, b are constants. If it is x

known that the system has only one stable equilibrium configuration, the possible values of a and b are a) a = 1, b = 2

12.

b) a = 1, b = -2

c) a = -1, b = 2

d) a = -1, b = -2

The acceleration of the 1kg block immediately after the string is cut is (g = 10ms-2).

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a) 4ms-2

13.

b) 4.1ms-2

c) 16ms-2

d) 40ms-2

A circular pan with its side wall inclined inward at 530 as shown is rotating about its central vertical axis with a constant angular velocity. A ball placed at the edge rotates along with it as shown. If the ball exerts a force of 22.5 N on the side wall and 23.5N on the bottom surface, the mass of the ball is

a) 1kg

14.

c) 3kg

4kg

Acceleration of 10kg block, when system released from rest, is

a) 0.5ms-2

15.

b) 2kg

Case(A)

Case (B)

b) 1 ms-2

c) 1.25 ms-2

d) 1.66ms-2

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The reaction force between the 5kg and 6kg block in case A and case B will be a) equal and non zero

b) unequal with case A being more

c) unequal with case B being more

d) equal and zero

For Question No. 16 to 20: Each question consider of two statements: one is Assertion (A) and the other is Reason(R). You are to examine these two statements and select the answer using the code given below. a) Both A and R individually correct, but R is the correct explanation of A b) Both A and R individually correct, but R is the not correct explanation of A c) A is true but R is false d) A is false but R is true 16.

Assertion (A): If a block is released from rest, when reaches the bottom point of the wedge, its speed is same irrespective whether the wedge is fixed or the wedge is fee to move. Reason(R): Mechanical energy is conserved in both cases.

17.

Assertion (A): A body is at rest on floor. You lift it vertically up and bring it to rest at a point h above the ground. The work done by you is zero. Reason(R): Any non-zero work done by a force on a body results in change in kinetic energy of the body.

18.

Assertion (A): The negative of the work done by the conservative internal forces on s system equals to change in its potential energy. Reason(R): Work energy theorem.

19.

Assertion (A): In a tug of war that team wins which applies more tension force on string then their opponents. Reason(R): The winning team must be having stronger players.

20.

Assertion (A): If a block starts moving at t = 0 with an acceleration ‘a’ then its kinetic energy at time t with respect to two non-inertial frames which have same acceleration in a direction is not always same. Reason(R): The work energy theorem ∆ K − ∆ W is not for non-inertial frame.

Study Material

Match the following Questions 21 to 25 21.

22.

Two columns are given in each question. Match the elements of Column-I with Column-II. Column-I

Column-II

i) Friction force

p) Contact force

ii) Normal reaction

q) Electromagnetic force

iii) Tension in a string

r) Gravitational force

iv) Force between two charges of mass m

s) Nuclear force

a) (i-p,q), (ii-p,q), (iii-q,s), (iv-q,r)

b) (i-p,q), (ii-p,q), (iii-q), (iv-q,r)

c) (i-p,q), (ii-p), (iii-q,s), (iv-q)

d) (i-q), (ii-p), (iii-q,s), (iv-p,q,r)

In Column-I there are some motions of a body and Column-II contains the list of concepts that can be used for the analysis of these motions Column-I

Column-II

i) A body moving in a vertical circle

p) Conservation of energy

ii) A body moving in a horizontal circle

q) Conservation of momentum

iii) A body dropped from a height on a block attached to the top of a vertical spring iv) Rocket propulsion

23.

r) Centripetal force s) Centrifugal force

a) (i-p,q,r), (ii-q,r,s), (iii-p,q,s), (iv-p)

b) (i-p,s), (ii-p,r), (iii-q), (iv-p,q,r)

c) (i-p,r), (ii-q,s), (iii-p), (iv-p,q)

d) (i-p,r,s), (ii-r,s), (iii-p,q), (iv-q)

When two bodies collide they come in contact at t = t1 and loses contact at t = t2. This (t2-t1) is a very small time interval. Consider this small time interval and match the following. i) Elastic collision

p) Kinetic energy decreases and potential energy increases and then potential energy decreases and kinetic energy increases

ii) Inelastic collision

q) Kinetic energy + potential energy is conserved

iii) Perfectly inelastic collision

r) Momentum is conserved

iv)Oblique elastic collision

s) Kinetic energy decreases and potential energy increases and then situation remains same

a) (i-q,r), (ii-p,r), (iii-p,q,r), (iv-q,r)

b) (i-p,r), (ii-q,r), (iii-q,r), (iv-p,r)

c) (i-p,q,r), (ii-p,q,r), (iii-q,r,s), (iv-p,q,r)

d) (i-r,s), (ii-r,s), (iii-r,s), (iv-r,s)

Study Material 24.

For each of the following four cases of Column-I match the range of force F in column B so that the block m is not slipping on the surface with which it has contact. For all cases m = 2kg; µ = 0.2 Take g = 10m/s2 Column-I

Column-II

p) (48N, 80N)

500   500 N, N 11   19

q) 

380   220 N, N 11   19

r) 

 220 380  N, N 17   23

s) 

25.

a) (i-p), (ii-q), (iii-q,r), (iv-p,q,r,s)

b) (i-p), (ii-q), (iii-r,s), (iv-q,r)

c) (i-q), (ii-p,s), (iii-q,s), (iv-r,s)

d) (i-q,r), (ii-p,q), (iii-r), (iv-p,q,r)

A body is moving in a vertical circle of radius R, considering motion from top point of circle bottom most point of circle, match the following. Column-I

Column-II

i) Tangential accelertaion p) Always increases ii) Centripetal acceleration

q) Always decreases

iii) Angular velocity

r) First increases then decreases

Study Material iv) Potential energy

s) Variable depending on angle made with vertical by string t) Variable independent of the made angle with vertical by string

a) (i-r,s), (ii-q,r), (iii-p,t), (iv-p,q)

b) (i-q,s), (ii-r,s), (iii-p,r), (iv-q,r)

c) (i-p,s), (ii-q,r), (iii-q,s), (iv-p,r)

d) (i-r,s), (ii-q,s), (iii-p,s), (iv-p,s)

Write the final answer to each question in this section in the column provided. 26.

The block ABCDE of mass 5m has BC part spherical, of radius 1m, is on frictionless horizontal surface. BC is quarter circle. A small mass m is released at B and slides down. How far away from D does it hit the floor? (g = 10ms-2)

27.

The kinetic energy of a particle of mass 1kg moving along a circle of radius 1m depends on time t as K = t4. Find the force acting on the particle as function of t.

28.

Spring is already in a compressed position with initial compression = 5cm. The 10kg block is allowed to fall. Determine the maximum compression of the spring before the block rebounds. (g = 10ms-2).

29.

A long plank of mass M = 8kg, length l = 1m rests on a horizontal surface. Coefficient of friction µ1 = 0.2 . A small block, mass m = 2kg rests on the right extreme and of M, on the rough top surface. At t = 0, a force F=

Study Material 25.5N is applied on M towards the right. (see figure). The block m does not slide on M. At t = 3s, force F is increased to 31N. The block m falls off M’s surface at t =7s. Determine the coefficient of friction between m and M.

30.

A particle is released on the smooth inside wall of a cylindrical tank at A with a velocity u which makes an angle α with the horizontal tangent. When the particle reaches a point B, a distance h below A, determine the angle β (as an inverse function of cosine) made by its velocity with the horizontal tangent at B.

Passage-I (31 to 33): Two blocks of masses 10kg and 5kg are placed on a rough horizontal floor as shoen in figure. The strings and pulley are light and pulley is frectionless. The coefficient of friction between 10kg block and surface is 0.3 while that between 5kg block and surface is 0.2. A time varying horizontal force. P=5t Newton (t is in sec) in applied on 5kg block as known. [Take g=10ms-2]

31.

The motion of block starts at t = t0, then t0 is a) 14s

32.

b) 8s

c) 9s

The friction force between 10kg block and surface at t = a) zero and 10N

b) 10N and 35N

d) 12s

t0 is in between 2

c) 12.5N and 17.5N

d) 12.5N certain value

Study Material 33.

The acceleration of 5kg block at t = 2t0 is a) 12 ms-2

b)

14 ms − 2 5

c)

14 −2 ms 9

d) 2 ms-2

Passage-II (34 to 36): A block of mass 4kg is pressed against a rough wall by two perpendicular horizontal force F1 and F2 as shown into figure, coefficient of static friction between the block and floor is 0.6 and that of kinetic friction is 0.5. [Take g=10ms-2]

34.

For F1 = 300N and F2 = 100N, find the direction and magnitude of friction force acting on the block. a) 180N, vertically upwards

b) 40N, vertically upwards

2 5

c) 107.7N, making an angle of tan −1   with the horizontal in upward direction.

2 5

d) 91.6N, making an angle of tan −1   with the horizontal in upward direction.

35.

For F1=150N and F2=100N, find the direction and magnitude of friction force action on block.

2 5

a) 90N, making an angle of tan −1   with the horizontal in upwards direction

2 5

b) 75N, making an angle of tan −1   with the horizontal in upwards direction

2 5

c) 107.7N, making an angle of tan −1   with the horizontal in upwards direction d) Zero

36.

For data of Question No.35, find the magnitude of acceleration of block. a) Zero

b) 22.5 ms-2

c) 26.925 ms-2 d) 8.175 ms-2

Study Material 37.

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass 2m and m just after the string is cut will be

a)

g upwards, g downwards 2

b) g upwards,

c) g upwards, 2g downwards

38.

g downwards 2

c) 2g upwards, g downwards

For the situation shown in the figure, the block is stationary wrt incline fixed in an elevator. The elevator is having an acceleration of

5 a0 whose components are shown in the figure. The surface is rough and

coefficient of static friction between the incline and block is µ S . Determine the magnitude of force exerted by incline on the block. [Take a0 =

a)

39.

mg 10

b)

g and θ = 37 0 , µ S = 0.6 ] 2

9 mg 25

c)

3 mg × 41 25

d)

13 mg 2

For the situation shown in the figure, mass of block A in equilibrium is [Assume all the pulley as light and frictionless and string as inextensible and light]

Study Material

a)

40.

2 kg sin θ

b) 1 kg

c) 18 kg

d)

54 sin θ kg 5

If coefficient of friction between all surfaces (for the shown diagram) is 0.4, then find the minimum force F to have equilibrium of the system. [Take g = 10 m/s2]

a) 62.5N

41.

b) 150 N

c) 135 N

d) 50 N

If the acceleration of wedge in the shown arrangement is a m/s2 towards left, then at this instant acceleration of the block (magnitude only) would be

Study Material

a) 4a m / s 2

42.

b) a 17 − 8 cos α m / s 2

c)

17 a m / s 2

d)

17 cos

α × a m / s2 2

Seven pulleys are connected with the help of 3 light strings as shown in the figure below. Consider P3, P4, P5 as light pulleys and pulleys P6 and P7 have masses m each. For this arrangement mark the correct statement(s)

a) Tension in the string connecting P1, P3 and P4 is zero. b) Tension in the string connecting P1, P3 and P4 is

mg . 3

c) Tension in all the 3 strings are same and equal to zero. d) Acceleration of P6 is g downward and that of P7 is g upward Passage (43 to 45): A system of two blocks is placed on a rough horizontal surface as shown in the figure below. The coefficient of static and kinetic friction at two surfaces are shown. A force F is horizontally applied on the upper block as shown. Let f1, f2 represent the frictional forces between upper and lower surfaces of contact, respectively and a1, a2 represent the acceleration of 3kg and 2kg block, respectively.

Study Material

43.

If F is gradually increasing force then which of the following statement(s) would be true? a) For a particular value of F(>m) moving at a speed 4m/s as shown in the figure. How far the body slides along the platform? [Take µ = 0.2 and g = 10 m / s 2 ]

a) 8m

64.

b) 12m

c) 6m

d) 4m

Two blocks both having masses 5kg are connected by a light string as shown in the figure. A horizontal force of 30N is applied on one of the blocks, the coefficient of friction between the blocks and surface is 0.5. Under these circumstances, the string is vibrating in its fundamental mode in resonance with a tuning fork of frequency 100Hz. Length of string is 50cm and its linear mass density is 1g/m. The friction forces between A and surface and between are, respectively

a) 25N and 25N

65.

b) 10N and 25N

c) 20N and 10N

d) 10N and 20N

A light rod can rotate freely in vertical plane about the hinge at its bottom end. Two strings tie the top of the rod with blocks A and B as shown. At the instant shown in figure, if block A is moving down with speed 3 m/s, then the speed of B would be

Study Material

a) 6 m/s

b)

18 m/s 5

c)

24 m/ s 5

d) None of these

Matrix-Match Type Question: 66.

A block of mass m is released from rest on a smooth inclined plane (wedge), the wedge itself is resting on smooth horizontal surface as shown in the figure. Assume that the block undergoes a vertical displacement of h. For this situation, match the entries of Column-I with the entries of Column-II.

Column-I

Column-II

A) Work done by normal reaction force on the block, is

P) Positive

B) Work done by normal reaction force exerted by block on the wedge is

Q) Negative

C) Total work done by normal reaction force (acting between block and wedge) on block and wedge is D) Total work done by all forces on block is

67.

R) Zero

S) Less than mgh in magnitude

The figure shows a pulley-mass system which is kept in an elevator moving up with acceleration g. Then the tension T in the string is given by

Study Material

a)

68.

m1m2 g m1 + m2

b)

2m1m2 g m1 + m2

c)

4m1m2 g m1 + m2

d)

(m1 − m2 )g m1 + m2

A car of mass m is driven with acceleration a along a straight level road against a constant external resistive force F. When the velocity of car is v, the rate at which the engine of the car is doing work will be a) Fv

69.

b) mav

c) (F + ma)v

d) (ma – F) v

A car starts from rest and attains a kinetic energy K by accelerating without slipping along a horizontal road in 20s. If air resistance is neglected, the work done by external forces, which accelerates the car will be a) Zero

70.

b) K

c)

K 20

d)

K 10

A boy of mass m is sliding down a vertical pole by pressing it with a horizontal force F. If µ is the coefficient of friction between his palms and the pole, the acceleration with which he slide down, is a) g

71.

b)

µF m

c) g +

µF m

d) g −

µF m

A truck moving on a smooth horizontal surface with a uniform speed u is carrying dust. If a mass ∆ m of the dust leaks from the truck in vertical downward direction in time ∆ t , the force needed keep the truck moving at its constant speed, is a) Zero

72.

b)

u∆m ∆t

c) ∆ m ×

du dt

d) u

∆m du + ∆m × ∆t dt

A body of mass 1kg is dropped from a height h = 40cm on a horizontal platform fixed to one end of a vertical spring as shown in figure. The other end of the spring is fixed to ground. As a result of the impact of ball with platform, the spring compresses by x = 10cm. [Neglect the mass of platform. The force constant of spring is [g = 10 m/s2]. Assume that during collision to energy is lost.]

Study Material

a) 600 N/m

73.

b) 1000 N/m

c) 300 N/m

d) 5000 N/m

A spring is compressed between blocks of masses m1 and m2 placed on a smooth horizontal surface as shown in figure. When the blocks are released, they have initial velocity of v1 an v2 in the direction shown. The blocks travel distances x1 and x2 before coming to rest. The ratio of

a)

74.

m1 m2

b)

m1 m2

c)

x1 is x2

m2 m1

d)

m2 m1

A uniform rod of mass M and length L is resting between a rough wall and a rough floor as shown in the figure. The coefficient of friction between any two surfaces is µ . The angle of θ for which the rod is just on the verge of slipping is

1− µ 2    µ 

a) tan −1 

1− µ 2 

 b) tan −1   2µ 

1− µ 2 

 c) sin −1   2µ 

d) None of these

Study Material

75.

A heavy spherical ball is constrained in a frame as shown in figure. All the surfaces are frictionless, the frame is moving with limiting acceleration so that ball is just on the verge of leaving contact with the frame. For this situation (limiting condition), mark out the correct statement(s).

a) If α > β , then ball will leave the contact with frame first from left side. b) If α > β , then ball will leave the contact with frame first from right side c) The ball leave the contact with frame first from left side whatever be the relation between

α and β . d) The ball leave the contact with frame first from right side whatever be the relation between

α and β . 76.

Sand is falling from a stationary hopper on a flat cart being pulled by a constant force. The rate at which sand is falling on cart is constant. The horizontal component of force exerted by the falling sand on the cart is

a) Continuously increasing

b) Continuously decreasing

c) Constant

d) First increasing and then decreasing

Passage Questions (77 to 79): A thin uniform rod of mass m and length L is hinged at one end from other end a light string is attached. The string is wound over a frictionless pulley (having mass 2m) and a block of mass 2m is connected to string on other side of pulley as shown. The system is released from rest when the rod is making an angle of 370 with horizontal. Based on above information answer the following questions:

Study Material

77.

Just after release of the system from rest, acceleration of block is a)

78.

72g , downwards 121

48g , downwards 119

c)

90g , downwards 121

d)

90g upwards 121

Just after release of the system, the resultant force exerted by hinge on rod is a) 0.7 mg

79.

b)

b) 0.92 mg

c) 0.53 mg

d) mg

Just after release of the system from rest, the resultant force exerted by hinge H2 on pulley is a)

46 mg in upward direction 121

b)

46 mg in downward direction 121

c)

438 mg in upward direction 121

c)

438 mg in downward direction 121

Matrix-Match Type Question: 80.

In Column-I four statements corresponding to different situations are given. In Column-I statements two vector quantities are compared in last, while comparison is given in Column-II. Match the entries of Column-I with the entries of Column-II. Column-I

Column-II

A) Two blocks of masses 2kg and 4kg are placed one over the other on a rough horizontal surface as shown in figure. The coefficient of friction between blocks is 0.2 while between 4kg block and surface is 0.1. An horizontal force of 5N is applied to 2kg block. Then friction force experienced

→  BA  and friction force experience by B due to  

by B due to A f

P) Same in magnitude

Study Material

→  BS  are  

surface B f

B) A particle is performing SHM about x = 0, it crosses a position

Q) Different in magnitude

x = x0 at t = t1 and at t = t2, there is no other crossing at x = x0 →

→

between t2 – t1. If v 1 and v 2 are velocity of particle at t1 and →

→

t2 respectively, then v 1 and v 2 are

C) Two points A and B are marked on the axis of a uniformly

R) Same in direction

charged ring at same distance from centre on two sides as →

→

shown. If E 1 and E 2 are electric field intensities at A and →

→

B respectively, then E 1 and E 2 are

→

D) A single turn carrying current of 1A is considered. If A

S) Opposite in direction

→

is its area vector and M is it corresponding magnetic moment, →

→

then A and M are

81.

A block of mass 10kg is in equilibrium as shown in the figure. If the block is displaced in vertical direction by a small amount, then angular frequency of resulting motion is

Study Material

a) 10 2

82.

b) 10 5

d) 5 5

c) 5 2

A 5kg ring attached to a spring slides along a smooth fixed rod which is inclined at an angle of 600 with horizontal, as shown in figure. The spring is in its relaxed state when ring is at A, the natural length of spring is 0.5m. The value of spring constant k, so that ring is having zero velocity at B is

a) 500 N/m

b) 323 N/m

c) 484 N/m

d) 286 N/m

Only One Correct Option Questions: →

83.

^

^

In the figure shown the acceleration of A is, a A = 15 i + 15 j then the acceleration of B is: (A remains in contact with B)

84.

^

^

^

^

a) 6 i

b) -15 i

c) -10 i

d) -5 i

Two blocks A and B of masses m & 2m respectively are held at rest such that the spring is in natural length. Find out the accelerations of both the blocks just after release.

Study Material

a) g ↓, g ↓

85.

b)

g g ↓, ↑ 3 3

c) 0, 0

d) g ↓, 0

A bob is hanging over a pulley inside a car through a string. The second end of the string is in the hand of a person standing in the car. The car is moving with constant acceleration ‘a’ directed horizontally as shown in figure. Other end of the string is pulled with constant acceleration ‘a’ vertically. The tension in the string is equal to

a) m g 2 + a 2

86.

b) m g 2 + a 2 − ma

d) m( g + a )

c) m g 2 + a 2 + ma

Two blocks ‘A’ and ‘B’ each of mass ‘m’ are placed on a horizontal surface. Two horizontal force F and 2F are applied on both the blocks ‘A’ and ‘B’ respectively as shown in figure. The block A does not slide on block B. Then the normal reaction acting between the blocks is:

a) F

b) F/2

c)

F 3

d) 3F

Study Material 87.

In the arrangement shown, by what acceleration the boy must go up so that 100kg block remains stationary on the wedge. The wedge is fixed and friction is absent everywhere.

a) 2 m/s2

88.

b) 4 m/s2

c) 6 m/s2

(Take g = 10 m/s2)

d) 8 m/s2

A system is shown in the figure. A man standing on the block is pulling the rope. Velocity of the point of string in contact with the hand of the man is 2m/s downwards. The velocity of the block will be: [Assume that the block does not rotate]

a) 3 m/s

89.

b) 2 m/s

c) 1/2 m/s

d) 1 m/s

In the figure shown the velocity of lift is 2m/s while string is winding on the motor shaft with velocity 2m/s and block A is moving downwards with a velocity of 2m/s, then find out the velocity of block B.

a) 2 m/s ↑ 90.

b) 2 m/s ↓

c) 4 m/s ↑

d) None of these

System is shown in the figure. Assume that cylinder remains in contact with the two wedges. The velocity of cylinder is

Study Material

a)

91.

19 − 4 3

u m/ s 2

13u m/ s 2

b)

3u m/ s

c)

d)

7 u m/s

The beads A and B move along a semicircular wire frame as shown in figure. The beads are connected by an inelastic

string

which

always

remains

tight.

At

an

instant

the

speed

of

A

is

u,

∠BAC = 45 0 and ∠BOC = 75 0 , where O is the centre of the semicircular are. The speed of bead B at that instant is:

a)

92.

2u

b) u

c)

u

d)

2 2

2 u 3

A plank is held at an angle α to the horizontal (Fig.) on two fixed supports A and B. The plank can slide against the supports (without friction) because of its weight Mg. Acceleration and direction in which a man of mass m should move so that the plank does not move.

 

m  down the incline M

b) g sin α 1 +

 

m  up the incline M

d) g sin α 1 +

a) g sin α 1 + c) g sin α 1 +

 

M  down the incline m

 

M  up the incline m

Study Material

93.

Objects A and B each of mass m are connected by light inextensible cord. They are constrained to move on a frictionless ring in vertical plane as shown in figure. The objects are releases from rest at the positions shown. The tension in the cord just after release will be

a) mg 2

94.

b)

mg 2

c)

mg 2

d)

mg 4

A pendulum of mass m hangs from a support fixed to a trolley. The direction of the string when the trolley rolls up a plane of inclination α with acceleration a0 is

a) θ = tan −1 α

95.

 a0   g

b) θ = tan −1 

g   a  0

c) θ = tan −1 

 a0 + g sin α    g cosα 

d) θ = tan −1 

The acceleration of the block B in the above figure, assuming the surfaces and the pulleys P1 and P2 are all smooth.

a)

F 4m

b)

F 6m

c)

F 2m

d)

3F 17 m

Study Material

96.

In the figure shown all block are of equal mass ‘m’. All surfaces are smooth. The acceleration of the block A with respect ground.

a)

4 g sin θ 1 + 3 sin 2 θ

b)

4 g sin 2 θ 1 + 3 sin 2 θ

c)

4 g sin θ

d) None of these

1 + 3 sin 2 θ

Match the Column Questions: 97.

Column-I gives four different situations involving two blocks of mass m1 and m2 placed in different ways on a smooth horizontal surface as shown. In each of the situations horizontal forces F1 and F2 are applied on blocks of mass m1 and m2 respectively and also m2F1 < m1 F2. Match the statements in Column-I with corresponding results in Column-II. Column-I

Column-II

A) Both the blocks are connected by massless inelastic string. The magnitude of tension in the string is

p)

m1 m2  F1 F2   −  m1 + m2  m1 m2 

q)

m1 m2 m1 + m2

 F1 F2   +   m1 m2 

r)

m1 m2 m1 + m2

 F2 F1   +   m2 m1 

B) Both the blocks are connected by massless inelastic string. The magnitude of tension in the blocks is

C) The magnitude of normal reaction between the blocks is

D) The magnitude of normal reaction between the blocks is

 F +F 

2  s) m1 m2  1 + m m  1 2 

Study Material

98.

The system shown below is initially in equilibrium. Masses of the blocks A,B,C,d and E are respectively 3m, 3m, 2m 2m and 2m. Match the conditions in Column-I with the effects in Column-II.

Column-I

Column-II

A) After spring 2 is cut, tension in string AB

p) Increases

B) After spring 2 is cut, tension in string CD

q) Decreases

C) After spring between C and pulley is cut, tension in string AB

r) Remain constant

D) After spring between C and pulley is cut, tension in string CD

s) Zero

Subjective Questions 99.

In the figure shown, blocks A and B move with velocities v1 and v2 along horizontal direction. Find the ratio of

v1 . v2

100.

Find the acceleration of B.

Study Material

101.

Find the acceleration of A w.r.t. ground

102.

Find the acceleration of C w.r.t. ground

Only One Correct Option Questions: 103.

In the arrangement shown in the figure mass of the block B and A are 2m, 8m respectively. Surface between B and floor is smooth. The block B is connected to block C by means of a pulley. If the whole system is released then the minimum value of mass of the block C so that the block A remains stationary with respect to B is: (Co-efficient of friction between A and B is µ .)

Study Material

a)

104.

m µ

b)

2m µ +1

c)

10m 1− µ

d)

10 m µ −1

Two block of masses m1 and m2 are connected with a massless unscratched spring and placed over a plank moving with an acceleration ‘a’ as shown in figure. The coefficient of friction between the blocks and platform is µ .

a) Spring will be stretched if a > µ g b) Spring will be compressed if a ≤ µ g c) Spring will neither be compressed nor be stretched for a ≤ µ g d) Spring will be in its natural length under all conditions

105.

A bead of mass m is located on a parabolic wire with its axis vertical and vertex directed towards downward as in figure and whose equation is x2 =ay. If the coefficient of friction is µ , the highest distance above the x-axis at which the particle will be in equilibrium is

a) µ a

b) µ 2 a

c)

1 2 µ a 4

d)

1 µa 2

Study Material 106.

In the shown arrangement if f1, f2 and T be the frictional forces on 2kg block, 3kg block & tension in the string respectively, then their values are:

a) 2N, 6N, 3.2N

b) 2N, 6N, 0N

c) 1N, 6N, 2N

d) Data insufficient to calculate the required values.

107.

A block of mass 2kg is given a push horizontally and then the block starts sliding over a horizontal plane. The figure shows the velocity-time graph of the motion. The co-efficient of sliding friction between the plane and the block is: (Take g = 10m/s2)

a) 0.02

108.

b) 0.20

c) 0.04

d) 0.40

The coefficient of friction between 4kg and 5kg blocks is 0.2and between 5kg block and ground is 0.1 respectively. Choose the correct statements

a) Minimum force needed to cause system to move is 17N b) When force is 4N static friction at all surfaces is 4N to keep system at rest c) Maximum acceleration of 4kg block is 2m/s2 d) Slipping between 4kg and 5kg blocks start when F is 17N

109.

The two blocks, m = 10kg and M = 50kg are free to move as shown. The coefficient of static friction between the blocks is. 5 and there is no friction between M and the ground. A minimum horizontal force F is applied to hold m against M that is equal to.

Study Material

a) 100 N

110.

b) 50 N

c) 240 N

d) 180 N

The coefficient of friction between block of mass m and 2m is µ = 2 tan θ . There is no friction between block of mass 2m and inclined plane. The maximum amplitude of two block system for which there is no relative motion between both the blocks.

a) g sin θ

111.

k m

b)

mg sin θ k

c)

3mg sin θ k

d) None of these

A force F =t is applied to a block A as shown in figure, where t is time in second. The force is applied at t =0 seconds when the system was at rest. Which of the following graph correctly gives the frictional force between A and horizontal surface as function of time t. [Assume that at t = 0, tension in the string connecting the two blocks is zero].

a)

112.

b)

c)

d)

A block of mass m is attached with a massless spring of force constant k. The block is placed over a fixed rough inclined surface for which the block of mass M required to move the block up the plane is (neglect mass of string and pulley and friction in pulley.)

Study Material

a)

113.

3 m 5

b)

4 m 5

c)

6 m 5

d)

3 m 2

A plank of mass M1 = 8kg with a bar of mass M2 = 2kg placed on its rough surface, lie on a smooth floor of elevator ascending with an acceleration g/4. The coefficient of friction is µ = 1 / 5 between m1 and m2. A horizontal force F = 30N is applied to the plank. Then the acceleration of bar and the plank in the reference frame of elevator are:

a) 3.5 m / s 2 , 5 m / s 2

114.

b) 5 m / s 2 ,

50 m / s2 8

c) 2.5 m / s 2 ,

25 m / s2 8

d) 4.5 m / s 2 , 4.5 m / s 2

A block of mass 1kg lies on a horizontal surface in a truck. The coefficient of static friction between the block and the surface is 0.6 if the acceleration of the truck is 5m/s2, the frictional force acting on the block is: a) 5 N

115.

b) 6 N

c) 10 N

d) 15 N

Two blocks with masses M1 and M2 of 10kg respectively are placed as in figure. µ S = 0.2 between all surfaces, tension in string and acceleration of M2 block will be

Study Material

a) 250 N, 3 m/s2

b) 200 N, 6m/s2

c) 306 N, 4.7 m/s2

d) 400 N, 6.5 m/s2

May have More than One options Correct: 116.

The value of mass m for which the 100kg block remains is static equilibrium is

a) 35kg

117.

b) 37kg

c) 83kg

d) 85kg

A block A (5kg) rests over another block B (3 kg) placed over a smooth horizontal surface. There is friction between A and B. A horizontal force F1 gradually increasing from zero to a maximum is applied to A so that the blocks move together without relative motion. Instead of this another horizontal force F2, gradually increasing from zero to a maximum is applied to B so that the blocks move together without relative motion. Then

a) F1 (max) = F2(max) 118.

b) F1 (max) > F2(max) c) F1 (max) < F2(max)

d) F1 (max):F2(max)=5.3

Two blocks of same mass m = 10kg are placed on rough horizontal surface as shown in figure. Initially tension

π  t  is applied as shown 6 

in the massless string is zero and string is horizontal. A horizontal force F = 40 sin 

Study Material on the block A for a time interval t = 0 to t = 6sec. Here F is in Newton and between block B and ground is 0.30. (Take g = 10m/sec2). Match the statements in Column-I with the time intervals (in seconds) in Column-II.

Column-I

Column-II

A) Friction force between block B and ground is zero in the time interval

p) 0 < t < 1

B) Tension in the string is non zero in the time interval

q) 1 < t < 3

C) Acceleration of block A is zero in the time interval

r) 3 < t < 5

D) Magnitude of friction force between A and ground is decreasing in the time interval

119.

s) 5 < t < 6

Two blocks of mass m and 2m are slowly just placed in contact with each other on a rough fixed inclined plane as shown. Initially both the blocks are at rest on inclined plane. The coefficient of friction between either block and inclined surface is µ . There is no friction between both the blocks. Neglect the tendency of rotation of blocks on the inclined surface. Column-I gives four situation. Column-II gives condition under which statements in Column-I are true. Match the statement in column-I with corresponding conditions in Column-II.

Column-I

Column-II

A) The magnitude of acceleration of both blocks are same if

p) µ = 0

B) The normal reaction between both the blocks is zero

q) µ > 0

C) The net reaction exerted by inclined surface on each block make same angle with inclined surface on block D) The net reaction exerted by inclined surface on block of mass 2m is double that of net reaction exerted by inclined

r) µ > tan θ

Study Material s) µ < tan θ

surface on block of mass m if

120.

A block of mass m is put on a rough inclined plane of inclination θ , and is tied with a light thread shown. Inclination θ is increased gradually form θ = 00 to θ = 900. Match the Columns according to corresponding curve.

Column-I

Column-II

A) Tension in the thread versus θ

B) Normal reaction between the block and the incline versus θ

p)

q)

C) Friction force between the block and the incline versus θ

D) Net interaction force between the block and the incline versus θ

r)

s)

Study Material

121.

A smooth ring A of mass m can slide on a fixed horizontal rod. A string tied to the ring passes over a fixed pulley B and carries a block C of mass M as shown in figure. At an instant the string between the ring and the pulley makes an angle α with the rod. What acceleration will the ring start moving if the system is released from rest with α = θ ?

122.

Two blocks of masses 5kg and 7kg are connected by uniform rope of mass 4kg as shown in the figure. An upward force F = 200N is applied on the system. Calculate

(a) Acceleration of the system

(b) Tension at the top of the heavy rope

(c) tension at the mid point of the rope (Take g = 10m/s2)

123.

An aerostat of mass m starts coming down with a constant acceleration a. Determine the ballast mass to be dumped for the aerostat to reach the upward acceleration of the same magnitude. The air drag is to be neglected.

Study Material 124.

The masses of the bodies A and B in figure are 20kg and 10kg, respectively. They are initially at rest on the floor and are connected by a weightless string passing over a weightless and frictionless pulley. An upward force F is applied to the pulley. Find the acceleration a1 of body A and a2 of body B when F is

(a) 98 N

125.

(b) 196 N

(c) 394 N

Two masses M1 = M2 = M are arranged as shown in the figure. Find the acceleration of mass M2. The pulleys are massless and frictionless.

126.

A block of mass 50kg is released from the position of rest for which the spring is under a tension of 45N. If the pulley are massless and the block goes 150mm down, find the velocity of the block at this instant.

127.

In figure M=1.9kg, m = 0.1kg, θ = cos −1 (0.8) , the wedge M is moving frictionlessly on the floor. Find the distance moved by M where m reaches the floor starting from a height of 2m.

Study Material

128.

A chain of length l is placed on a smooth spherical surface of radius R with one of ends fixed at the top of the sphere. What will be the acceleration of the each element of the chain when its upper end is released? It is

π R  .  2 

assumed that the length of the chain l < 

129.

Block A in figure weighs 4N and block B weights 8N. The coefficient of sliding friction between all surfaces is 0.25. Find the force P necessary to drag block B to the left at constant speed. (a) If rests A on B and moves with it.

(b) If A is held at rest

(c) If A and B are connected by a light flexible cord passing around a fixed frictionless pulley.

130.

In figure A is 10kg block and B is a 5kg block, (a) determine the minimum mass of C which must be placed on A to prevent it from sliding. If coefficient of static friction µ S between A and the table is 0.25 (b) the block C is suddenly lifted off A, what is the acceleration of block A, if the coefficient of kinetic friction between A and the table is 0.20?

Study Material

131.

In the arrangement the masses of bodies are equal to m0, m1, and m2, the masses of the pulley and the threads are negligible, and there is no friction in the pulley. Find the acceleration ‘a’ with which the body m0 comes down, and the tension of the thread binding together the bodies m1 and m2 if the coefficient of friction between the bodies and the horizontal surface is equal to µ . Consider possible cases.

132.

In the figure coefficient of kinetic friction between the block 3kg and 2kg is µ = 0.3 . The horizontal table surface is smooth. Find (i) The acceleration of mass

133.

(ii) Tension in the strings

A small body A starts sliding down from the top of a wedge whose base is equal to 2.10m. The coefficient of friction between the body and the wedge surface is µ = 0 . 140. At what value of angle will the time of sliding be the least? What will it be equal to?

134.

Consider a chain of length L lying on a horizontal table top such that a portion of length x hangs vertically (see figure). It is assumed that at x = 0, the velocity of the chain was zero. Determine the velocity v of the chain as a function of the overhang length x.

Study Material

135.

A cylinder of mass m and radius r rests on two supports of the same height (see figure). One support is stationary, while the other slides from under the cylinder at a velocity v. Determine the force of normal pressure N exerted by the cylinder on the stationary support at the moment when the distance between the points A and B of the supports is AB = r 2 , assuming that the supports where very close to each other at the initial instant. The friction between the cylinder and the supports should be neglected.

136.

A cylinder and a wedge with a vertical face, touching each other, move along two smooth inclined planes forming the same angle α with the horizontal (see figure). The masses of the cylinder and the wedge are m1 and m2 respectively. Determine the force of normal pressure N exerted by the wedge on the cylinder, neglecting the friction between them.

137.

ABCD is square; along the sides AB, CB, DC, and DA forces act equal respectively to 6, 5, 8, and 12N. Find the algebraic sum of their moments about the centre ‘O’ of the square, if the side of the square is 4m.

Study Material

138.

ABCD is square; along the sides AB, BC, DC, and DA act forces equal to 1, 9, 5 and 3N. Find force, passing through the centre of the square, and the couple which are together equivalent to the given system.

139.

Two identical, uniform, frictionless spheres, each of weight W, rest in a rigid rectangular container as shown in figure, find, in terms of W, the forces acting on the spheres due to (a) the container surfaces and (b) one another, if the line of centres of the spheres makes angle of 450 with the horizontal.

140.

Find the pull P required to lift the load W shown in figure assuming the efficiency of the system is 78%.

Study Material

141.

A heavy homogeneous sphere is suspended by a light string, one end of which is attached to a vertical wall and the other, to a point on the vertical line through the centre of sphere as shown in figure. What should be the coefficient of friction between the sphere and the wall for the sphere to remain in equilibrium?

142.

A short, right circular cylinder of weight W rests in a horizontal, V-shaped notch of angle 2α as shown in figure. If the coefficient of friction is µ , find the horizontal force parallel to the axis necessary for slipping to occur.

143.

A rectangular block 1m wide and 2m high is dragged to the right along a level road at a constant speed by a horizontal force P, as shown in figure. The coefficient of sliding friction is 0.4, the block weighs 50kg and it cg is at centre. (a) Find the force required (b) find the value of h at which the block just start to tip (c) find the line of action of the normal force N exerted on the block by the surface, if the height h is 1m.

Study Material

144.

A garage door is mounted on an overhead rail, as in figure. The wheels at A and B have rusted so that they do not roll but slide along the track. The coefficient of sliding friction is 0.5. The distance between the wheels is 2m, and each is 0.5m in from the vertical sides of the door. The door is symmetrical and weighs 800N. It is pushed to left at constant velocity by a horizontal force P? (a) If the distance h is 1.5, what is the vertical component of the force exerted on each wheel by the track? (b) Find the maximum value of h can have without causing one wheel to leave the track.

145.

In the arrangement shown in figure the person of weight W pulls himself up with an acceleration a. If the weight of the chair is w. then find the tension in the rope. Is it always possible to lift the chair?

Study Material 146.

In figure the block M1 has mass 1kg and the trolley M2 is 10kg. Friction between M 1 and M2 is 0.5, while the trolley moves frictionlessly on the floor. If θ = 30 0 , determine the maximum F for which there is no slippage between M1 and M2.

147.

In figure m1 = 1kg and m2 = 2kg. The pulley is movable. At t =0, both masses touch the ground and the string is tant. A vertically upward, time-dependent force F=2t, where t is the time in second and F is the force in Newton, is applied to the pulley. Calculate (a) the time when m1 is lifted off the ground and (b) m2 is lifted off the ground (c) when t = 25s, what is the acceleration of the masses as seen by an observer outside.

148.

In the system shown in figure, M=13.4kg, m1=1kg, m2 = 2kg and m3 = 3.6kg. The coefficient of static friction between m1 and m2 is 0.75 and that between m2 and M is 0.6. All other surfaces are frictionless (a) What minimum horizontal force F must be applied to M so that m2 does not slip on M? (b) What is the maximum value of F for which m1 does not slip on m2? (c) If F = 100N, does m1 slip with respect to m2 or does m2 slip with respect to M?

Study Material

149.

A noose is placed around a log and used to pull it with a force F. How will the tension of the ropes forming the loop depend on the magnitude of the angle α ? In what conditions will the tension of the rope in the sections AB and AC be larger than in the section AD?

150.

One end of a heavy uniform board of weight P and length l presser against a corner between a wall and a floor. A rope is attached to the other end of the board (figure). Determine the tension in the rope BC if the angle between the board and the rope is β = 90 0 . How will the tension in the rope change as the angle α between the board and the floor increases if the angle β remains constant?

151.

A ladder of weight P and length l is placed against a smooth vertical wall at an angle α = 30 0 . The centre of gravity of the ladder is at a height h from the floor (figure). A man pulls the ladder at its middle point in a horizontal direction with a force F. What is the minimum value of F which will permit the man to detach the upper end of the ladder from the wall? The friction against the floor is such that the bottom of the ladder does not slide.

Study Material

152.

A wooden block lies on an inclined surface (figure). With what force should this block be pressed against the surface to retain it in equilibrium? The weight of the block is P = 2kgf, the length of the inclined surface l =1m and h = 60cm. The coefficient of friction of the block against the inclined surface is µ = 0.4 .

153.

A heavy log is pulled on an inclined surface with the aid of two parallel ropes secured as shown in figure. The weight of the log is P =400kg f, the height of the inclined surface h = 1m, and its length l =2m. What force F should be applied to each rope to pull the log up?

Study Material

KEY 1)

c

2)

d

3)

c

4)

a

5)

d

6)

b

7)

c

8)

c

9)

b

10)

b

11)

c

12)

d

13)

a

14)

d

15)

c

16)

d

17)

d

18)

c

19)

d

20)

c

21)

b

22)

d

23)

c

24)

a

25)

d

27)

β 2 + t6 N

28)

5 2

29)

µ 2 = 0.1

30)

  −1  u cos α cos  2 gh  1+ 2 u 

26)

24 m 5

( 17 − 1)cm

31)

a

32)

c

33)

c

34)

c

35)

b

36)

d

37)

a

38)

d

39)

b

40)

a

41)

b

42)

a,c

43)

d

44)

c

45)

b

46)

(A-R),

47)

48)

49)

50)

(B-R), (C-Q,R) (D-P,Q,R) 51)

d

52)

b

53)

c

54)

c

55)

56)

b

57)

d

58)

d

59)

d

60)

c

61)

b

62)

c

63)

d

64)

d

65)

c

66)

(A-P,S)

67)

c

68)

c

69)

a

70)

d

(B-P,S) (C-Q,S) (D-Q,S) 71)

a

72)

b

73)

c

74)

b

75)

d

76)

c

77)

a

78)

c

79)

c

80)

(A-P,S) (B-P,S) (C-P,S) (D-P,R)

81)

a

82)

b

     

Study Material

Laws of Motion & Equilibrium Answers

(W + w)(a + g ) ; only for 2W > w

121)

Mg cos θ m + M cos 2 θ

145)

122)

(a) 2.5m/s2 (b) 112.5 N (c) 87.5 N

146)2.52N

123)

∆m =

2ma (g + a )

3g

147) (a) 9.81s,

(b) 19.62s,

(c) 15.19ms2, 2.69 m/s2. (b) 0, 0

(c) 0.05m/s2, 9.9m/s2

124)

(a) 0, 0

148) (a) 117.7N, (b) 147N, (c) m2slips on M

125)

2g m / s2 5

149) T =

126)

1.28 m/s

150) T =

127)

1.66m.

151) F =

128)

a=

129)

(a) 3.00N (b) 4.00N (c)5.00N

130)

MC = 10kg, a = 1.96 m/s2

131)

 m − µ (m1 + m2 )   m0 m2 g (1 + µ )   a= 0  g ; T =   m0 + m1 + m2   m0 + m1 + m2 

132)

(i) 5.75 m/s2

133)

T = 1.05s, α = 49 0

134)

v=

135)

N=

136)

 2m1 m2    tan α m + m  1 2 

137)

10N-m.

F 2 cos

Rg [1 − cos(l / R )] l

(ii) T1 =17.38N, T2 = 40.5N

g x L mg mv 2 − 2 2r

α 2

, α > 120 0

P cos α 2 2 P h sin α × l cos 2 α

152) F = 1.4 kgf. 153) F =

Ph = 50 kgf. 4l

Study Material 138)

6 2 N inclined at 450 to the side AB.

139)

(a) Bottom: 2W; side; W

140)

1923.08N

141)

µ ≥1

142)

µW sin α

143)

(a) 196N

144)

(a) 100N, 700N

(b)

2W

(b) h=1.25m (c) 0.4m to the right (b) 2.0m