Newton's Laws, Force & Friction-291

By N.K.C. SIR NEWTONS

LAW FORCE & FRICTION

KEY CONCEPT ............................................................. Page –2 OBJECTIVE QUESTION BANK...................................... Page –4 SUBJECTIVE LEVEL - I .. .............. .............................. Page –11 SUBJECTIVE LEVEL - II . .............. .............................. Page –16 ANSWER KEY ............................................................... Page –19

KEY CONCEPT FORCE There are, basically, five forces, which are commonly encountered in mechanics. Weight : Weight of an object is the force with which earth attracts it. It is also called the force of gravity or the gravitational force.

1. (i)

(ii)

Contact Force : When two bodies come in contact they exert forces on each other that is called contact forces. (a) Normal force (N) : It is the component of contact force normal to the surface. It measures how strongly the surfaces in contact are pressed together. (b) Frictional force : It is the component of contact force parallel to the surface. It opposes the relative motion (or attempted motion) of the two surfaces in contact.

(iii)

Tension : The force exerted by the end of a taut string, rope or chain is called the tension. The direction of tension is to pull the body while that of normal reaction is to push the body.

(iv)

Spring force : The force exerted by a spring is given by F = – kx, where x is the change in length and k is the stiffness constant or spring constant (units Nm–1).

2.

NEWTON'S LAWS Newton's First Law : Every particle continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by the action of an applied force.

3.

Newton's Second Law :

4.

Newton's Third Law : Whenever two bodies interact they exert forces on each other which are equal in magnitude and opposite in direction. So whenever body A exerts a force F on body B, B exerts a force – F on A.

Fnet

ma

Inertial Reference Frame : A reference frame in which Newton’s first law is valid is called an inertial reference frame. An inertial frame is either at rest or moving with uniform velocity. Non-Inertial Frame : An accelerated frame of reference is called a non-inertial frame. Objects in noninertial frames do not obey Newton’s first law. Pseudo Force : It is an imaginary force which is recognized only by a non-inertial observer to explain the physical situation according to Newton’s law. The magnitude of this force FP is equal to the product of the mass m of the object and acceleration a of the frame of reference. The direction of the force is opposite to the direction of acceleration. FP = – ma The force of friction comes into action only when there is a relative motion between the two contact surfaces or when an attempt is made to have it. The force of friction on each body is in a direction opposite to its motion (existing or impending) relative to other body.

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5.

Static friction : The frictional force acting between any two surfaces at rest with respect to each other is called the force of static friction (fs). fs where

6.

sN

s is the static coefficient of friction.

Kinetic friction : The frictional force acting between surfaces in relative motion with respect to each other is called the force of kinetic friction or sliding friction (fk). fk = where

kN

k is the coefficient of kinetic friction. s

>

k

Angle of friction ( ) : Mathematically, the angle of friction ( ) may be defined as the angle between the normal reaction N and the resultant of the maximum friction force f and the normal reaction. f N Since f = N, therefore, tan =

Thus tan =

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OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Take approx. 2 minutes for answering each question. Q.1 A book is at rest on a table. What is the “reaction” force according to Newton's third law to the gravitational force by the earth on the book? (A) the normal force exerted by the table on the book (B) the normal force exerted by the table on the ground (C) the normal force exerted by the ground on the table (D) the gravitational force exerted on the earth by the book Ans.

Q.2

A weight can be hung in any of the following four ways by string of same type. In which case is the string most likely to break?

(A) A

(B) B

(C) C

(D) D

Ans.

Q.3

A rope of mass 5 kg is moving vertically in vertical position with an upwards force of 100 N acting at the upper end and a downwards force of 70 N acting at the lower end. The tension at midpoint of the rope is (A) 100 N (B) 85 N (C) 75 N (D) 105 N

Ans.

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Q.4

A stunt man jumps his car over a crater as shown (neglect air resistance) (A) during the whole flight the driver experiences weightlessness (B) during the whole flight the driver never experiences weightlessness (C) during the whole flight the driver experiences weightlessness only at the highest point (D) the apparent weight increases during upward journey

Ans.

Q.5

A flexible chain of weight W hangs between two fixed points A & B which are at the same horizontal level. The inclination of the chain with the horizontal at both the points of support is . What is the tension of the chain at the mid point? (A)

W . cosec 2

(B)

W . tan 2

(C)

W cot 2

(D) none

Ans.

Q.6

What should be the minimum force P to be applied to the string so that block of mass m just begins to move up the frictionless plane. Mg cos (A) Mg tan 2 (C) (B) Mg cot 2 (D) None 1 sin

Ans.

Q.7

A spring of force constant k is cut into two pieces such that one piece such that one piece is double the length of the other. Then the long piece will have a force constant of (A) (2/3) k (B) (3/2) k (C) 3k (D) 6k

Ans.

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Q.8

Find the acceleration of 3 kg mass when acceleration of 2 kg mass is 2 ms–2 as shown in figure. (A) 3 ms–2 (B) 2 ms–2 –2 (C) 0.5 ms (D) zero

Ans.

Q.9

Block of 1 kg is initially in equilibrium and is hanging by two identical springs A and B as shown in figures. If spring A is cut from lower point at t=0 then, find acceleration of block in ms–2 at t = 0. (A) 5 (B) 10 (C) 15 (D) 0

Ans.

Q.10

If the string & all the pulleys are ideal, acceleration of mass m is g 2 (C) g

(B) 0

(A)

(D) dependent on m

Ans.

Question No. 11 to 13 (3 questions) A particle of mass m is constrained to move on x-axis. A force F acts on the particle. F always points toward the position labeled E. For example, when the particle is to the left of E, F points to the right. The magnitude of F is a constant F except at point E where it is zero. The system is horizontal. F is the net force acting on the particle. The particle is displaced a distance A towards left from the equilibrium position E and released from rest at t = 0. Q.11

What is the period of the motion? (A) 4

2Am F

(B) 2

2Am F

(C)

2Am F

(D) None

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Ans.

Q.12

Velocity – time graph of the particle is

(A)

(B)

(C)

(D)

Ans.

Q.13

Find minimum time it will take to reach from x = – (A)

3 mA ( 2 1) (B) 2 F

mA ( 2 1) F

A to 0. 2

(C) 2

mA ( 2 1) F

(D) None

Ans.

Q.14

Two blocks are connected by a spring. The combination is suspended, at rest, from a string attatched to the ceiling, as shown in the figure. The string breaks suddenly. Immediately after the string breaks, what is the initial downward acceleration of the upper block of mass 2m ? (A) 0 (B) 3g/2 (C) g (D) 2g

Ans.

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Q.15

A block is projected upwards on an inclined plane of inclination 37° along the line of greatest slope of = 0.5 with velocity of 5 m/s. The block 1st stops at a distance of __________ from starting point (A) 1.25 m (B) 2.5 m (C) 10 m (D) 12.5 m

Ans.

Q.16

A block of mass 2 kg slides down an incline plane of inclination 30°. The coefficient of friction between block and plane is 0.5. The contact force between block and plank is : (A) 20 Nt

(B) 10 3 Nt

(C) 5 7 Nt

(D) 5 15 Nt

Ans.

Q.17

A body is placed on a rough inclined plane of inclination . As the angle is increased from 0° to 90° the contact force between the block and the plane (A) remains constant (B) first remains constant then decreases (C) first decreases then increases (D) first increases then decreases

Ans.

Q.18

A force F (A) – ˆi (C) – 2.4 ˆi

ˆi 4ˆj acts on block shown. The force of friction acting on the block is : (B) – 1.8 ˆi

(D) – 3 ˆi

Ans.

Q.19

Block B of mass 100 kg rests on a rough surface of friction coefficient = 1/3. A rope is tied to block B as shown in figure. The maximum acceleration with which boy A of 25 kg can climbs on rope without making block move is : 4g g g 3g (A) (B) (C) (D) 3 3 2 4 ETOOS Academy Ltd. : F-106, Road no.2, Indraprastha Industrial Area, End of Evergreen Motors (Mahindra Showroom), BSNL Office Lane, Jhalawar Road, Kota, Rajasthan (324005) [8]

Ans.

Q.20

A block placed on a rough inclined plane of inclination ( =30°) can just be pushed upwards by applying a force "F" as shown. If the angle of inclination of the inclined plane is increased to ( = 60°), the same block can just be prevented from sliding down by application of a force of same magnitude. The coefficient of friction between the block and the inclined plane is (A)

3 1 3 1

(B)

2 3 1 3 1

(C)

3 1 3 1

(D) None of these

Ans.

Q.21

With what minimum velocity should block be projected from left end A towards end B such that it reaches the other end B of conveyer belt moving with constant velocity v. Friction coefficient between block and belt is . (A)

gL

(C)

3 gL

(B) (D) 2

2 gL gL

Ans.

For Q. 22 to Q.26 refer figure-1.(5 questions) Q.22 When F = 2N, the frictional force between 5 kg block and ground is (A) 2N (B) 0 (C) 8 N (D) 10 N Ans.

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Q.23

When F = 2N, the frictional force between 10 kg block and 5 kg block is (A) 2N (B) 15 N (C) 10 N (D) None

Ans.

Q.24

The maximum "F" which will not cause motion of any of the blocks. (A) 10 N (B) 15 N (C) data insufficient

(D) None

The maximum acceleration of 5 kg block (A) 1 m/s2 (B) 3 m/s2

(D) None

Ans.

Q.25

(C) 0

Ans.

Q.26

The acceleration of 10 kg block when F = 30N (B) 3 m/s2 (C) 1 m/s2 (A) 2 m/s2

(D) None

Ans.

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(NEWTONS LAW FORCE & FRICTION) SUBJECTIVE LEVEL - I Q.1

Figure shows three blocks in contact and kept on a smooth horizontal surface. What is ratio of force exerted by block A on B to that of B on C.

Ans.

Q.2

A force F appli ed to an object of mass m 1 produces an acceleration of 3.00 m/s2. The same force

applied to a second object of mass m2 produces an acceleration of 1.00 m/s2. (a) What is the value of the ratio m1 / m2 ? (b) If m1 and m2 are combined, find their acceleration under the action of the force F.

Ans.

Q.3

Two forces, F1 = (– 6i – 4j) N and F2 = ( – 3i + 7j) N, act on a particle of mass 2.00 kg that is initially at rest at coordinates (– 2.00 m, + 4.00 m). (a) What are the components of the particle's velocity at t = 10.0 s? (b) In what direction is the particle moving at t = 10.0s? (c) What displacement does the particle undergo during the first 10.0s? (d) what are the coordinates of the particle at t = 10.0 s?

Ans.

Q.4

In the system shown in figure, a horizontal force of magnitude Fx acts on the 8.00 kg object. The horizontal surface is frictionless. (a) For what values of Fx does the 2.00 kg object accelerate upward? (b) For what values of Fx is the tension in the cord zero? (c) Plot the acceleration of the 8.00 kg object versus Fx. Include values of Fx from – 100 N to + 100 N.

Ans.

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Q.5

A van accelerates down a hill (Fig.), going from rest to 30.0 m/s in 6.00 s. During the acceleration, a toy (m = 0.100 kg) hangs by a string from the van’s ceiling. The acceleration is such that the string remains perpendicular to the ceiling. Determine (a) the angle and (b) the tension in the string.

Ans.

Q.6

Block A of mass m/2 is connected to one end of light rope which passes over a pulley as shown in the Fig. Man of mass m climbs the other end of rope with a relative acceleration of g/6 with respect to rope. Find acceleration of block A and tension in the rope.

Ans.

Q.7

(a) (b)

To point the side of a building, painter normally hoists himself up by pulling on the rope A as in figure. The painter and platform together weigh 200N. The rope B can withstand 300N. Find the maximum acceleration of the painter. tension in rope A (i) when painter is at rest (ii) when painter moves up with an acceleration 2 m/s2.

Ans.

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Q.8

Same spring is attached with 2 kg, 3 kg and 1 kg blocks in three different cases as shown in figure. If x1, x2 and x3 be the constan extensions in the spring in these three cases then find the ratio of their extensions.

(a)

(b)

(c)

Ans.

Q.9

What horizontal force must be applied to the cart shown in figure in order that the blocks remain stationary relative to the cart? Assume all surfaces, wheels, and pulley are frictionless.

Ans.

Q.10

Inclined plane is moved towards right with an acceleration of 5 ms–2 as shown in figure. Find force in newton which block of mass 5 kg exerts on the incline plane. (All surfaces are smooth)

Ans.

Q.11

Find force in newton which mass A exerts on mass B if B is moving towards right with 3 ms–2. Also find mass of A. (All surfaces are smooth)

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Ans.

Q.12 Force F is applied on upper pulley. If F = 30t where t is time in second. Find the time when m1 loses contact with floor. Ans.

Q.13

Consider a large truck carrying a heavy load, such as steel beam. A significant hazard for the driver is that the load may slide forward, crushing the cab, if the truck stops suddenly in an accident or even in braking. Assume, for example, that a 10000 kg load is located on the flat bed of a truck moving at 12.0 m/s. Assume the load is not tied down to the truck and assume that the coefficient of friction between the load and the truck bed is 0.500. Calculate the minimum stopping distance for the truck for which the load will not slide forward relative to the truck.

Ans.

Q.14

In the figure, what should be mass m so that block A slide up with a constant velocity?

Ans.

Q.15

A block of mass 1 kg is horizontally thrown with a velocity of 10 m/s on a stationary long plank of mass 2 kg whose surface has a = 0.5. Plank rests on frictionless surface. Find the time when m1 comes to rest w.r.t. plank.

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Ans.

Q.16 Block M slides down on frictionless incline as shown. Find the minimum friction coefficient so that m does not slide with respect to M. Ans.

Q.17 The coefficient of static and kinetic friction between the two blocks and also between the lower block and the grou nd are = 0.6 and k = 0.4. Find the value of tension T applied on the lower s block at which the upper block begins to slip relative to lower block. Ans.

Q.18

Find the acceleration of the blocks and magnitude & direction of frictional force between block A and table, if block A is pulled towards left witha force of 50N.

Ans.

Q.19

Coefficient of friction between 5 kg and 10 kg block is 0.5. If friction between them is 20 N. What is the value of force being applied on 5 kg. The floor is frictionless

Ans.

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SUBJECTIVE LEVEL - II Q.1

(a) (b) (c)

The diagram shows particles A and B, of masses 0.2 kg and m kg respectively, connected by a light inextensible string which passes over a fixed smooth peg. The system is released from rest, with B at a height of 0.25 m above the floor. B descends, hitting the floor 0.5 s later. All resistances to motion may be ignored. Find the acceleration of B as it descends. Find the tension in the string while B is descending and find also the value of m. When B hits the floor it comes to rest immediately, and the string becomes slack. Find the length of time for which B remains at rest on the ground before being jerked into motion again.

Ans.

Q.2

A 1kg block ‘B’ rests as shown on a bracket ‘A’ of same mass. Constant forces F1 = 20N and F2 = 8N start to act at time t = 0 when the distance of block B from pulley is 50cm. Time when block B reaches the pulley is _________.

Ans.

Q.3

Two men of masses m 1 and m2 hold on the opposite ends of a rope

passing over a frictionless pulley. The mass m1 climbs up the rope with an acceleration of 1.2 m/s2 relative to the rope. The man m2 climbs up the rope with an acceleration of 2.0 m/s2 relative to the rope. Find the tension in the rope if m1 = 40 kg and m2 = 60 kg. Also find the time after which they will be at same horizontal level if they start from rest and are initially separated by 5m. Ans.

Q.4

In figure shown, pulleys are ideal m1 > 2 m2. Initially the system is in equilibrium and string connecting m2 to rigid support below is cut. Find the initial acceleration of m2?

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Q.5

(a) (b) (c)

The system shown adjacent is in equilibrium. Find the acceleration of the blocks A, B & C all of equal masses m at the instant when (Assume springs to be ideal) The spring between ceiling & A is cut. The string (inextensible) between A & B is cut. The spring between B & C is cut. Also find the tension in the string when the system is at rest and in the above 3 cases.

Ans.

Q.6

In the system shown. Find the initial acceleration of the wedge of mass 5M. The pulleys are ideal and the cords are inextensible. (there is no friction anywhere).

Ans.

Q.7 (a) (b)

A block of mass m lies on wedge of mass M as shown in figure. Answer following parts separately. M With what minimum acceleration must the wedge be moved towards right horizontally so that block m falls freely. Find the minimum friction coefficient required between wedge M and ground so that it does not move while block m slips down on it.

Ans.

Q.8

A car begins to move at time t = 0 and then accelerates along a straight track with a speed given by V(t) = 2t2 ms–1 for 0 < t < 2 After the end of acceleration, the car continues to move at a constant speed. A small block initially at rest on the floor of the car begins to slip at t = 1sec. and stops slipping at t = 3 sec. Find the coefficient of static and kinetic friction between the block and the floor.

Ans.

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Q.9

An inclined plane makes an angle 30° with the horizontal. A groove OA = 5 m cut in the plane makes an angle 30° with OX. A short smooth cylinder is free to slide down the influence of gravity. Find the time taken by the cylinder to reach from A to O. ( g = 10 m/s2)

Ans.

Q.10

A thin rod of length 1 m is fixed in a vertical position inside a train, which is moving horizontally with constant acceleration 4 m/s2. A bead can slide on the rod, and friction coefficient between them is 1/2. If the bead is released from rest at the top of the rod, find the time when it will reach at the bottom.

Ans.

Q.11

Three identical rigid circular cylinders A, B and C are arranged on smooth inclined surfaces as shown in figure. Find the least value of that prevent the arrangement from collapse.

Ans.

Q.12

A block of mass 50 kg resting on a horizontal surface is acted upon by a force F which varies as shown in the figure. If the coefficient of friction between the block and surface is 0.2, find the time (in second) when the block will come to rest.

Ans.

Q.13

Find minimum normal force to be applied by each hand to hold three identical books in vertical position. Each book has mass 'm' and value of coefficient of friction between the books as well as between hand and the book is .

Ans.

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OBJECTIVE QUESTION BANK ONLY ONE OPTION IS CORRECT. Q.1

D

Q.2

C

Q.3

B

Q.4

A

Q.5

C

Q.6

A

Q.7

B

Q.8

B

Q.9

A

Q.10

C

Q.11

A

Q.12 A

Q.13

B

Q.14

B

Q.15 A

Q.16

D

Q.17 B

Q.18 A

Q.19

B

Q.20

C

Q.21

B

Q.22 A

Q.23 A

Q.24

A

Q.25

C

Q.26

A

(NEWTONS LAW FORCE & FRICTION) SUBJECTIVE LEVEL - I Q.2

m1 (a) m 2

1 2 3 (b) a = 3/4 m/s ]

Q.1

3:1

Q.3

(a) 45 m/s along negative x-axis and 15 m/s along positive y-axis 1 from positive x-axis 3

(b) = tan–1

Q.4

(a) Fx > 19.6 N

Q.5

(a) 30° (b)

Q.7 Q.9

(b) Fx

– 78.4 N

3 2

(c) ( 225 ˆi 75 ˆj) m

(d) ( 227 ˆi 79 ˆj) m

(c)

4g 13mg ,T= 9 18

Q.6

a=

(a) 5m/s2, (b)(i)100 N, (ii) 120 N

Q.8

x2 > x1 > x3 x1 : x2 : x3 : 15 : 18 : 10

m2 (M + m1 + m2) m g 1

Q.10

55

Q.11 5N, 16/31 kg

Q.14

1 kg

Q.15 4/3 sec

Q.18

10 ˆi

Q.19

Q.12 2 sec

Q.13

14.4m

Q.16 3/4

Q.17 40 N

30 N

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SUBJECTIVE LEVEL - II Q.1

(a) 2 ms–2, (b) 2.4 N, 0.3 (c) 0.2 s

Q.2

Q.3

556.8 N, 1.47 sec

Q.4

Q.5

(a) aA=

0.5 sec m1 2m 2 g 2m 2

3g =aB; aC=0; T=mg/2; 2

(b) aA= 2g , aB = 2g , ac = 0, T = 0; (c) aA= aB = g/2 , ac = g , T =

Q.6

Q.8

Q.12

2g/23

s

=0.4 ,

3.60

k

= 0.3

Q.9

2 sec

Q.13

N=

3mg ; T = 2mg 2

Q.7

(a) a = g cot , (b) mmin =

Q.10

1/2 sec

Q.11

m sin cos m cos2 M

tan–1

1 3 3

3mg 2

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