Work Power Energy Etoos

WORK POWER ENERGY

Contents Topic

Page No.

Theory

01 - 02

Exercise - 1

03 - 12

Exercise - 2

12 - 19

Exercise - 3

20 - 23

Exercise - 4

24 - 25

Answer Key

26 - 27

Syllabus Kinetic and potential energy ; Work and power ; Conservation of linear momentum and mechanical energy.

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WORK POWER ENERGY Work (W) : The work W done by a constant force F when its point of application undergoes a displacement s is defined as W = F.s = Fs cos  where  is the angle between F and s.Work is a scalar quantity and its SI units is N-m or joule (J).

Note:

1.

Only the component (F cos ) of the force F which is along the displacement contributes to the work done.

Fx ˆi  Fy ˆj  Fz kˆ and s  xˆi  yˆj  zkˆ

If

F=

then

W = F ·s = Fxx + Fyy + Fz z

 

Work done by a Variable Force : When the magnitude and direction of a force varies with position, The work done by such a force for an infinitesimal displacement ds is given by





dW = F · d s

In terms of rectangular components, XB

WAB 

YB

 Fx dx   Fydy   Fz dz

XA

2.

ZB

YA

ZA

Work Done by a Spring Force : The work done by the spring force for a displacement from xi to xf is given by

1 Ws   k x f2  x i2 2



3.



Work Energy theorem : Work done on a body can produce a change in its kinetic energy. Work is required to produce motion and it is also required to destroy motion. W = K = Kf – Ki

4.

Conservative Force : The force which does work in complete independence of the path followed the body is called a conservative force. The gravitational force, spring force and electrostatic force are the examples of conservative forces.

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5.

Non-Conservative Force : The work done by a non-conservative force not only depends on the initial and final positions but also on the path followed. The common examples of such forces are : frictional force and drag force of fluids.

6.

Potential Energy : The potential energy is defined only for conservative forces. B

UB–UA = – Fc .ds



A

7.

Conservative force : dU Fc = – dx At equilibrium,

dU =0 dx

The point B is the position of stable equilibrium, because

d 2U dx 2

The point C is the position of unstable equilibrium, because

>0

d 2U dx 2

tOB

(B) tAO < tOB

(C) tAO = tOB

(D) h = h

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PART - II : SUBJECTIVE QUESTIONS

1.

One end of a spring of natural length  and spring constant k is fixed at the ground and the other is fitted with a smooth ring of mass m which is allowed to slide on a horizontal rod fixed at a height  (figure). Initially, the spring makes an angle of  with the vertical when the system is released from rest. If the speed of the ring when the spring becomes vertical is (2 / 3)

2.

k m/s then find the value of angle  : m

A small block slides along a path that is without friction until the block reaches the section L = 3m, which begins at height h = 3m on a flat incline of angle 37°, as shown. In that section, the coefficient of kinetic friction is 0.50. The block passes through point A with a speed of

136 m/s. Find the speed

of the block as it passes through point B where the friction ends, in m/s. (Take g = 10 m/s 2)

3.

A particle A of mass

10 kg is moving in the positive direction of x. Its initial position is x = 0 & initial velocity 7

is 1 m/s. The velocity of particle at x = 10 is v m/s. Find value of v ? : (use the graph given)

Power (in watts)

4 2

10

x

(in m)

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4.

A ball is suspended from the top of a cart by a string of length 1 m. The cart and the ball are initially moving to the right at constant speed V, as shown in figure I. The cart comes to rest after colliding and sticking to a fixed bumper, as in figure II. The suspended ball swings through a maximum angle 60°. The initial speed is V =

2 K  1 m/s. Find value of K. (take g = 10 m/s )

V 60°

L

Figure (1)

Bumper

Figure (2)

Bumper

5.

A body of mass 1 kg is shifted from A to D on inclined planes by applying a force slowly such that the block is always is in contact with the plane surfaces. Neglecting the jerk experienced at points C and B, total work done by the applied force is 30K ? . Find K :

6.

A particle is projected vertically upwards with a speed of 16 m/s, after some time, when it again passes through the point of projection, its speed is found to be 8 m/s. It is known that the work done by air resistance is same during upward and downward motion. Then the maximum height ( in metre ) attained by the particle is ? (Take g = 10 m/s2 ) :

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PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS) *Marked Questions are having more than one correct option. 1.

A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed v, the electrical power output will be proportional to[JEE (Scr.) 2000, 3/105] (B) v2

(A) v 2.

(B)

(D)

(B) 2 Mg/k

(C) Mg/k

(D) Mg/2k

A particle moves under the influence of a force F = kx in one dimensions (k is a positive constant and x is the distance of the particle from the origin). Assume that the potential energy of the particle at the origin is zero, the schematic diagram of the potential energy U as a function of x is given by : [JEE(Scr) 2004, 3/84]

(A)

5.

(C)

An ideal spring with spring-constant k is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstreched. Then the maximum extension in the spring is : [JEE(Scr) 2002, 3/105] (A) 4 Mg/k

4.

(D) v4

A particle, which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as F(x) = –kx + ax3. Here k and a are positive constants. For x  0, the functional form of the potential energy U(x) of the particle is : [JEE(Scr) 2002, 3/105]

(A)

3.

(C) v3

(B)

STATEMENT - 1 :

(C)

(D)

[JEE 2007' 3/184] [JEE 2007, 3/81] [Conducted by Bombay]

A block of mass m starts moving on a rough horizontal surface with a velocity v. It stops due to friction between the block and the surface after moving through a certain distance. The surface is now tilted to an angle of 30º with the horizontal and the same block is made to go up on the surface with the same initial velocity v. The decrease in the mechanical energy in the second situation is smaller than that in the first situation. Because STATEMENT - 2 The coefficient of friction between the block and the surface decreases with the increase in the angle of inclination. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True.

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6.

A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4 k, respectively (see figure ). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure ) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio

y is x

[JEE 2008, 3/163] [conducted by IIT Roorkee]

Figure :

(A) 4

(B) 2

(C)

1 2

(D)

1 4

7.

A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. [ JEE 2009, 4/160, –1]

8.

A block of mass 2 kg is free to move along the x-axis. It is at rest and from t = 0 onwards it is subjected to a time-dependent force F (t) in the x direction. The force F (t) varies with t as shown in the figure. The kinetic energy of the block after 4.5 seconds is : [JEE' 2010, 8/163] conducted by IIT Madras] F(t)

N

4.5s

O

(A) 4.50 J

(B) 7.50 J

3s

(C) 5.06 J

t

(D) 14.06 J

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9.

10.

A block of mass 0.18 kg is attached to a spring of force-constant 2 N/m. The coefficient of friction between the block and the floor is 0.1. Initially the block is at rest and the spring is un-stretched. An impulse is given to the block as shown in the figure. The block slides a distance of 0.06 m and comes to rest for the first time. The initial velocity of the block in m/s is V = N/10. Then N is : [JEE' 2011, 4/160] [conducted by IIT Kanpur]

 x The work done on a particle of mass m by a force, K   x2  y2 



3/2

y

ˆi 

x



2

 y2

3/2



 ˆj  (K being a constant  

of appropriate dimensions), when the particle is taken from the point (a, 0) to the point (0, a) along a circular path of radius a about the origin in the x-y plane is : [JEE Advanced (P-1) 2013] (A)

11.

2K a

(B)

K a

(C)

K 2a

(D) 0

A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in ms–1) of the particle is zero, the speed (in ms–1) after 5 s is :

Paragraph for Questions 12 and 13 A small block of mass 1 kg is released from rest at the top of a rough track. The track is a circular are of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure below, is 150 J. (Take the acceleration due to gravity, g = 10ms–2) [JEE Advanced (P-2) 2013]

y

R 30º

P R

Q x

O

12.

The magnitude of the normal reaction that acts on the block at the point Q is: [JEE Advanced (P-2) 2013] (A) 7.5 N

13.

(B) 8.6 N

(C) 11.5 N

The speed of the block when it reaches the point Q is : (A) 5 ms–1

(B) 10 ms–1

(C)

10 3 ms–1

(D) 22.5 N [JEE Advanced (P-2) 2013] (D) 20 ms–1

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PART-II AIEEE (PREVIOUS YEARS PROBLEMS) * Marked Questions are having more than one correct option. 1.

If a body loses half of its velocity on penetrating 3 cm in a wooden block, then how much will it penetrate more before coming to rest? [AIEEE 2002, 4/300] (1) 1 cm (2) 2 cm (3) 3 cm (4) 4 cm

2.

A spring of force constant 800 N/m has an extension of 5cm. The work done in extending it from 5cm to 15cm is : [AIEEE 2002, 4/300] (1) 16 J (2) 8 J (3) 32 J (4) 24 J

3.

A spring of spring constant 5 × 103 N/m is stretched initially by 5 cm from the unstretched position. Then the work required to stretch it further by another 5 cm is : [AIEEE 2003, 4/300] (1) 12.50 N-m (2) 18.75 N-m (3) 25.00 N-m (4) 6.25 N-m

4.

A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time t is proportional to : [AIEEE 2003, 4/300] (1) t3/4 (2) t3/2 (3) t1/4 (4) t1/2

5.

A uniform chain of length 2 m is kept on a table such that a length of 60 cm hangs freely from the edge of the table. The total mass of the chain is 4 kg. What is the work done in pulling the entire chain on the table? [AIEEE 2004, 4/300] (1) 7.2 J (2) 3.6 J (3) 120 J (4) 1200 J   A force F  (5 ˆi  3ˆj  2kˆ ) N is applied over a particle which displaces it from origin to the point r  ( 2 ˆi  ˆj ) m .

6.

The work done on the particle in joules is : (1) – 7 (2) + 7 7.

(4) + 13

A body of mass m accelerates uniformly from rest to v1 in time t1. The instantaneous power delivered to the body as a function of time t is : [AIEEE 2004, 4/300] (1)

8.

[AIEEE 2004, 4/300] (3) + 10

m 1t t1

(2)

m12 t

(3)

t12

m1 t t1

(4)

m12 t t1

A body of mass m is accelerated uniformly from rest to a speed v in a time T. The instantaneous power delivered to the body as a function of time, is given by : [AIEEE 2005, 4/300] (1)

mv 2 T2

.t

(2)

mv 2 T2

.t2

(3)

1 mv 2 .t 2 T2

(4)

1 mv 2 2 .t 2 T2

9.

A particle of mass 100 g is thrown vertically upwards with a speed of 5 m/s. the work done by the force of gravity during the time the particle goes up is [AIEEE 2006, 1.5/180] (1) – 0.5 J (2) –1.25 J (3) +1.25 J (4) 0.5 J

10.

A ball of mass 0.2 kg is thrown vertically upwards by applying a force by hand. If the hand moves 0.2 m which applying the force and the ball goes upto 2 m height further, find the magnitude of the force. Consider g = 10 m/s 2 : [AIEEE 2006, 3/180] (1) 22 N (2) 4 N (3) 16 N (4) 20 N

11.

A particle is projected at 60° to the horizontal with a kinetic energy K. The kinetic energy at the highest point is : [AIEEE 2007, 3/120] (1) K (2) zero (3) K/4 (4) K/2

12.

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range [AIEEE 2008, 3/105] (1) 2 × 105 J – 3 × 105 J (2) 20,000 J – 50,000 J (3) 2,000 J – 5,000 J (4) 200 J – 500 J

13.

If two springs S1 and S2 of force constants k1 and k2, respectively, are streched by the same force, it is found that more work is done on spring S1 than on spring S2. [AIEEE 2012, 4/120] Statement 1 : If stretched by the same amount, work done on S1, will be more than that on S2. Statement 2 : k1 < k2 (1) Statement 1 is false, Statement 2 is true. (2) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1. (3) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1. (4) Statement 1 is true, Statmenet 2 is false.

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NCERT QUESTIONS 1.

The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative : (A) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket. (b) work done by gravitational force in the above case. (c) work done by friction on a body sliding down an inclined plane. (d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity, (e) work done by the resistive force of air on a vibrating pendulum in bringing into rest.

2.

A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction - 0.1. Compute the (a) work done by the applied force in 10 s, (b) work done by friction in 10 s, (c) work done by the net force on the body in 10 s (d) change in kinetic energy of the body in 10 s, and interpret your results.

3.

Given in figure are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

4.

The potential energy function for a particle executing linear simple harmonic motion is given by V(x) = kx2/2, where k is the force constant of the oscillator. For k = 0.5 N m–1, the graphj of V(x) versus x is shown in figure. Show that a particle of total energy 1 J moving under this potential must 'turn back' when it reaches x = ± 2 m. V(x)

x

5.

Answer the following : (a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained ? The rocket or the atmosphere ? (b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet's velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why ? (c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then as it comes closer and closer to the earth ? (d) In figure the man walks 2 m carrying a mass of 15 kg on his hands. In figure (ii) he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater ?

15kg

15kg

(i)

(ii)

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6.

A body is initially at rest. It undergoes on -dimensional motion with constant acceleration. The power delivered to it at time t is proportional to (i) t1/2 (ii) t (iii) t3/2 (iv) t2

7.

A body is moving uni-directionally under the influence of a source of constant power. Its displacement in time t is proportional to : (i) t1?2 (ii) t (iii) t3/2 (iv) t2

8.

A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by : F=  ˆi  2 ˆj  3 kˆ N where ˆi, ˆj, kˆ are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?

9.

An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton ? Obtain the ratio of their speeds. (Electron mass = 9.11 × 10–31 kg, proton mass = 1.67 × 10–27 kg, 1 eV = 1.60 × 10–19 J).

10.

A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey ? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is 10 m s–1 ?

11.

A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m3 in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump ?

12.

The bob of a pendulum is released from a horizontal position A as shown in figure. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point B, given that it dissipated 5% of its initial energy against air resistance ?

13.

A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the trolley's floor at the rate of 0.05 kg s–1. What is the speed of the trolley after the entire sand bag is empty ?

14.

A particle of mass 0.5 kg travels in a straight line with velocity v = ax3/2 where a = 5 m–1/2 s–1. What is the work done by the net force during its displacement from x = 0 x = 2m ?

15.

The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t ? (b) What is the kinetic energy of the air ? (c) Assume that the windmill converts 25% of the wind's energy into electrical energy, and that A = 30 m2, v = 36 km/h and the density of air is 1.2 kgm–2. What is the electrical power produced ?

16.

A person trying to loose weight (dieter) lifts a 10 kg mass 0.5 m, 1000 times. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much work does she do against the gravitational force ? (b) Fat supplies 3.8 × 107 J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up ?

17.

A large family uses 8 kW of power. (A) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW ? (b) Compare this area to that of the roof of a typical house.

18.

A bullet of mass 0.012 kg and horizontal speed 70 ms–1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.

19.

Two inclined frictionless tracks, one gradual and he other steep meet at A from where two stones are allowed to slide down from rest, one on each track (Figure). Will the stones reach the bottom at the same time ? Will they reach there with the same speed ? Explain. Given 1 = 30°, 2 = 60°, and h = 10 m, what are the speeds and times taken by the two stones ?

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Exercise # 1 PART-I A-1.*

(ACD) A-2.

(B)

A-3.

(C)

A-4.

(A)

B-1.

(D)

B-2.

(C)

B-3.

(C)

B-4.

(D)

B-5.*

(BD)

C-1.

(BD)

C-2.

(C)

C-3.

(C)

C-4.

(A)

C-5.

(B)

C-6.

(B)

C-7.

(C)

D-1.

(B)

D-2.

(B)

D-3.

(A)

D-4.

(B)

D-5.

(C)

D-6.

(A)

D-7.

(D)

D-8.

(C)

D-9.

(ACD) E-1.*

(ABC) E-2.*

(AD)

E-3.

(A)

E-4.

(D)

F-1.

(A)

F-2.

(D)

F-3.

(C)

F-4.

(B)

F-5.

(BC)

F-6.

(C)

F-7.

(C)

F-8.

(B)

F-9.*

(ABC) F-10.

(A)

PART-II 1.

(A)

2.

(B)

3.

(C)

4.

(C)

5.

(B)

6.

(D)

7.

(D)

8.

(B)

9.

(D)

10.

(D)

11.

(D)

12.

(C)

13.

(D)

14.

(B)

15.

(A) – p, r ; (B) – q, s ; (C) – q, r ; (D) – p

16.

(A) – t ; (B) – p ; (C) – s ; (D) – q

Exercise # 2 PART-I 1.

(A)

2.

(C)

3.

(C)

4.

(C)

5.

(D)

6.

(B)

7.

(C)

8.

(D)

9.

(B)

10.

(A)

11.

(D)

12.

(C)

13.

(D)

14.

(D)

15.

(D)

16.

(A)

17.

(A)

18.

(B)

19.

(D)

20.

(D)

21.

(C)

22.

(A)

23.

(AB)

24.

(AC)

25.

(ABD) 26.

29.

(AD)

(BCD) 27.

(BCD) 28.

5.

3

6.

8

(ABC)

PART-II 1.

53º 2.

4

3.

4

4.

9

Exercise # 3 PART-I 1.

(C)

2.

(D)

3.

(B)

4.

(A)

5.

(C)

6.

(C)

8.

(C)

9.

4

10.

(D)

11.

5

12.

(A)

13.

(B)

7.

8J

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PART-II 1.

(1)

2.

(2)

3.

(2)

4.

(2)

5.

(2)

6.

(2)

8.

(1)

9.

(2)

10.

(1)

11.

(3)

12.

(3)

13.

(1)

7.

(2)

Exercise # 4 1.

(a) +ve (b) –ve (c) – ve (d) +ve (e) – ve

2.

(a) 882 J (b) – 247 J; (c) 635 J (d) 535 J Work done by thenet force on a body equals change in its kinetic energy.

3.

(a) x > a ; 0 (c) x < a, x > b ; – V1 (b) –  < x <  ; V1 (d) – b/2 < x < –a/2, a/2 < x < b/2; –V1

5.

(a) rocket ; (b) For a conservative force work done over a path is minus of change in potential energy. Over a complete orbit, there is not change in potential energy; (c) K.E. increases, but P.E. decreases, and the sum decreases due to dissipation against friction; (d) in the second case.

6.

(b) t

7.

(c) t3/2

8.

12 J

9.

The electron is faster, ve/vp = 13.5

10.

0.082 J in each half; – 0.163 J

11.

43.6 W

12.

5.3 m s–1

13.

27 km h–1 (no change in speed)

14.

50 J

15.

(a) m =  Avt

(b) K =  A v3 t/2 (c) P = 4.5 kW h

16.

(a) 49,000 J

(b) 6.45 10–3 kg

17.

(a) 200 m2 (b) Comparable to the roof of a large house of dimension 14m × 14 m.

18.

21.2 cm, 28.5 J

19.

No, the stone on the steep plane reaches the bottom earlier; yes, they reach with the same speed v [mgh = (1/2) mv2] VB = VC = 14.1 m s–1, tB = 2 2 s, tC = 2 2 s

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