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## WAVE MOTION QUESTIONS AND ANSWERS FOR CLASS XI

Q1. If a body of mass 2kg. is at restand is hit by a mass of 4kg. moving with 3m/s, find fraction of the momentumretained by the moving body assuming the collision to be elastic and head-on.

Ans1. If n = m2/m1
(n – 1)/(1 + n) is the fraction of the momentum retained by the moving body so
(n – 1)/(1 + n) = (2 – 1)/(1 + 2) = 1/3.

Q2. If mass of the moving body ismuch greater than the mass of the body at rest than what is the approximatevilocity of the moving body after head-on collision?

Ans2. When the moving mass is muchgreater than the mass at rest then after the collision the heavier mass keepson moving with the same velocity and in the same direction.

Q3. At what point the potentialenergy of a body is taken to be zero?

Ans3. The potential energy of the bodyat the surface of the earth is taken to be zero (Potential energy = mgh, whereh is the height of the body from the surface.).

Q4. Does the work done on a body by aforce depend upon the path followed by it?

Ans4. May or may not be. If the force isconservative then it does not depend but if it is non-conservative (friction)then it depends.

Q5. If a body hits the ground from aheight h1 and rebounds to a height h2 after havinginelastic collision with the ground then what is the coefficient ofrestitution?

Ans5. e = Ö(h2/h1)

Q6. A body hits the ground with 50m/s velocity and has inelastic collision with the ground then with whatvelocity it will rebound if the coefficient of restitution is 0.2.

Ans6. Coefficient of restitution, e = v2/v1where v1 is the velocity with which the body hits the ground and v2is the velocity of rebound.
e = 0.2 = v2/50, so v2 = 10 m/sec.

Q7. A body at rest explodes in threefragments. Is it possible that two equal parts move in mutually perpendiculardirections with the same velocity and third mass moves midway between the two?

Ans7. No, it is not possible becausemomentum before the explosion is zero and after it also the momentum must bethe same. In the given situation the third particle must go in a directionopposite to the resultant of first two parts.

Q8. If the speed of a moving vehicleis increased by 200% then how much should be the change in the retarding forceto stop the vehicle over half the previous distance?

Ans8. (1/2)mv2 = F.S where Fis the retarding force and S is the distance over which the vehicle comes torest. When ‘v’ is increased by 200% then K.E. increases by 800%. As S is halvedthen F should be made 16 times.

Q9. If 20 Joules of work is done in compressinga spring from 0 cm to 6 cm then find the work done in compressing the same from3cm to 6 cm.

Ans9. W.D = (1/2) K(x22– x12). This is the work done when the spring iscompressed from x1 to x2.
20 = (1/2)K (36 – 0)
W = (1/2)K (36 – 9)
Solving the above equations we get W = 15 J.

Q1. A ball is dropped from rest at aheight of 20m. If it loses 30% of its kinetic energy on striking the ground,what is the height to which it bounces? How do you account for this loss inkinetic energy?

Ans1. Suppose the ball acquires avelocity ‘v’ after falling through a height of 20m.
Because the ball is dropped from rest, hence u = 0.
Hence, v2 = u2 + 2as
= 0 + (2 x 10 x 20) = 400
So, v = 20 m/s
Kinetic energy of the ball just before hiting the ground
= (1/2)mv2 = (1/2)m(400) = 200m Joule
Because the ball loses 30% of the kinetic energy on striking the ground, hencekinetic energy retained by the ball after striking the ground = 70% of 200m J
= 140m J
The energy loss is due to the inelastic collision with the ground.

Q2. Why is no energy being consumedin planetary motion.

Ans2. A planet is a heavenly body whichrevolves round the star (the sun). The force which is responsible for circularmotion, is called centripetal force. The direction of the centripetal force isalways towards the centre. Thus, the angle between force F and displacement Sis q = 90o at every point.Work done in moving planet, W = F.S = FSCosq = FS Cos90o.So, W = 0. Hence,no energyis being consumed in planetary motion.
height=82src=”./xi%20work,%20power%20and%20energy_files/image001.gif”alt=”workpowerandenergyf06q16i.gif (1336 bytes)” v:shapes=”_x0000_i1027″>

Q3. How will you justify thathydro-electric power plant is an illustration of law of conservation of energy?

Ans3. A hydroelectric power-plant isused in generating electric energy (Power).The potential energy of water storedat a height is converted into K.E. when water is made to rush down. This fallof water is used to rotate the turbine and the coil and armature of generatoris rotated and electricity is produced . Thus, the K.E. of the fall of water isconverted into electrical form of energy. Hence the hydroelectric power-plantis an example of law of conservation of energy.

Q4. If a body is dropped from aheight of 40m then after 3 inelastic collisions with the ground to which heightthe body will rise? (given: Coefficient of restitution = 0.5)

Ans4. If the body is dropped from aheight of ‘H’ and ‘e’ is the coefficient of restitution then after ‘n’inelastic collisions with the ground the body rises to a height ‘h’ given by
h = H.e2n.
h = 40 x (1/2)2 x 3= 40 x (1/26) = 40/64 = 0.625 m

Q5. A truck and a car are moving withthe same K.E. on a straight line road. If their engines are made off at thesame time, which one of them will stop at a lesser distance.

Ans5. Given : (1/2)m1v12= (1/2)m2v22 ………….(i)
If force of brakes be the same then m1a1 = m2a2……….(ii)
If truck stops over a distance S1 then v12 =2a1S1 ……..(iii)
If car stops over a distance S2 then v22 = 2a2S2………(iv)
From (i) and (ii)
(1/2)m1v12 = (1/2)m2v22………..(v)
From (ii) and (v)
v12/a1 = v22/a2…………(vi)
From (iii), (iv) and (vi)
2a1S1/a1 = 2a2S2/a2.
S1 = S2
Hence distances covered S1 and S2 are equal.

Q6. The power of a pump motor is 4KW. How much water  in kg/minute can it raise a height of 20m? (g = 10 m/s2)

Ans6. Given power of motor P = 4KW =4000 W
If mass of water raised in one second = m kg.
Total work done in lifting water,W = mgh
Power P = W/t, but t = 1 minute = 60 sec.
4000 = mgh/60
4000 = (m x 10 x 20)/60
m = 1200 kg.

Q7. A rod of length 3m is suspendedvertically from a fixed point. It is given an angular displacement of 60oin the vertical plane. If its mass per unit length is 2 kg then find the workdone?

Ans7. Let ‘m’ be the mass of the rod and’l’ be its length then
m = 2 x 3 = 6 kg
If the rod is displaced through an angle qthen the work done on it, W = mg(l/2)(1 – Cosq).
The effective length of the rod is taken to be (l/2) because in uniformdistribution of mass the centre of mass is at the geometric centre so
W = 6 x 10 x (3/2)(1 – Cos60) =45 J.

Q1. When a mass m2is at rest and mass m1 moving with velocity u1 hits itelastically, show that the fraction of the momentum transferred to the mass atrest is 2n/(1 + n) where n is ratio of the masses.

Ans1. v1 = (m1 -m2)u1/(m1 + m2).
v2 = 2m1u1/(m1 + m2).
This is from the theory of conservation of momentum.
Momentum of the mass m2 after collision,
P2 = m2v2 = (2m1m2u1)/(m1+ m2)
Fraction of momentum transferred to m2.
= (2m1m2u1)/(m1 + m2)m1u1= 2m2/(m1 + m2)
= 2n/(1 + n) ….[because m2/m1 = n]

Q2. Is it possible for work to bepositive negative or zero? "Explain with example.

Ans2. Work is a scalar quantity which isgiven by the scalar product of the force applied F and the displacement S movedby body i.e.
W = F.S,
W = FS Cos q ……….(1).
If q = 0. The work is maximum.
It remains +ve for the angle q between 0oto 90o
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or,
q lying between 270o and360o i.e., if  the displacement is in a directionoppisite to which the force is applied.
height=59src=”./xi%20work,%20power%20and%20energy_files/image003.gif”alt=”tutor2phyworkpowerandenergyf06q2i2.gif (1097 bytes)” v:shapes=”_x0000_i1038″>
Thus work is +ve if Cos q is +ve. The work done will be -veif Cosq is -ve i.e. q lies between 90o to 270o.
If
q = 90o then Cos 90o= 0.
Hence work done W = FS Cos 90o = 0.
Thus, W may be +ve, -ve or zero

Q3. The bob of a simple pendulum isreleased from a horizontal position. If the length of the pendulum is 2m, whatis the speed with which the bob arrives at the lowermost point? Given that itdissipates 10% of its initial energy against air resistance?

Ans3. Gravitational potential energy atthe highest position
= mg x 2 Joules = 2mg Joules
Kinetic energy at lowest position
= Potential energy at the highest position – the energy dissipitated againstair resistance or friction
= [mg x 2 – (10/100) x mg x2] Joule
= mg x 18/10 J
(1/2)mv2 = mg x18/10
or, v = 1.9 ms-1.

Q4. Give the various gravitationalunits of work. Give their relations also.

Ans4. We know that the physical workdone is given by
W = FS
(1) In S.I system,
If F = 1 kg weight or 1 kg force and S = 1m then,
W = (1 kg wt)(1m) = 1 kg m ……………(i).
Hence, one kgm is the gravitational unit of work in S.I (M.K.S) system and isdefined as the amount of work done if 1 kg force displaces a body through 1m inthe direction of the applied force.
(2) In C.G.S system,
F = 1 gmwt and S = 1cm,
W = (1 gm wt) (1 cm) = 1 gm cm ………………(ii).
Hence, one gm cm is the gravitational unit of work and is defined as the amountof work done, if 1 gm force displaces a body through 1 cm in the direction ofthe applied force.
1 gm cm = 980 ergs.
NOTE: 1 kg m = 9.8 Joules.

Q5. What is 1 e.v (electron volt)?Give its values in Joules. Give values of 1 MeV and 1 BeV also.

Ans5. One electron volt is the unit ofenergy in Atomic and Nuclear Physics.
One electron volt is the energy acquired by one elctron in moving it betweentwo point having a P.D of 1V.
Thus, 1eV = (1.6 x 10-19) C x1J/C = 1.6 x 10-19 Joules.
NOTE: The other practical units used are
1 Million electron volt = 1 MeV = 106 eV, 1 MeV = 106 x1.6 x 10-19 J, 1 MeV = 1.6 x10-13 Joules and
1 Billion eV = 109 eV, 1 BeV = 1.6 x 10-10joules.

Q6. When water is flowing through apipe then its velocity changes by 5%, find the change in the power of water?

Ans6. Power = Force xVelocity = Rate of change of momentum x velocity ={(mass/time) x velocity} xvelocity = {(adv) x v} x v =adv3 where ‘a’ is area of cross section, ‘d’ is the density of waterand ‘v’ is the velocity of flow of water.
Therefore, Power of water is directly proportional to the cube of velocity ofwater so let P = Kv3 (k is a constant and is equal  to ‘ad’.)Taking log on both sides
log P = 3log v + log k
Differentiating on both sides
DP/P = 3.Dv/v
percentage change in power,
DP/P x100 = 3 x 5%
= 15%.

Q7. The kinetic energy of rushing outwater from a dam is used in rotating a turbine. The pipe through which water isrushing is 2.4 meters and its speed is 12 m/sec. Assuming that whole of kineticenergy of the water is used in rotating the turbine, calculate the currentproduced if efficiency of the dynamo is 60% and the station transmits power at240 kV. Density of water = 103 kg/m3.

Ans7. Given that
r = radius of pipe = 1.2m, average speed of water v = 12 m/s
V = 240 kV = 240 x 103 volt, density ofwater p = 103 kg/m3.
Now, kinetic energy of rushing water per second i.e.
Power P = (1/2)(mass flowing per sec) x v2
= (1/2)pr2(l/t) rv2
= (1/2)
pr2 rv3
= (1/2) x 3.14 x (1.2)2x 103 x (12)3watt
= 3.9 x 106 watt
Current in the transmissioncables
current = output power/voltage
= (60% of power P)/(240 x 1000)
= [(60/100) x 3.9 x 106]/(240x 1000) = 9.75 amp.

Q1. A vehicle of mass m isaccelerated from rest when a constant power P is supplied by its engine; showthat :
(a) The velocity is given as a function of time by
v = (2Pt/m)1/2
(b) The position is given as a function of time by
s = (8P/9m)1/2t3/2.
(c) What is the shape of the graph between velocity and mass of the vehicle ifother factors remain same?
(d) What is the shape of the graph between displacement and power?

Ans1. (a) Given that Power = Fv = P =constant
i.e., m x (dv/dt) x v =P [as F = ma = m x (dv/dt)]
After rearranging and integrating on both sides
òv dv = ò(P/m) x dt
(v2/2) = (P/m) x t + C1
Now as initially the body is at rest, i.e., v = 0 at t = 0, so C1= 0.
v = (2Pt/m)1/2 …………(1)
(b) By definition v = (ds/dt),
Using eq (1) above,
ds/dt = (2Pt/m)1/2
On integration we get
òds = ò(2Pt/m)1/2 dt
s = (2P/m)1/2 x (2/3) xt3/2 + C2.
Now, as at t = 0, s = 0, so, C2 = 0
s = (8P/9m)1/2 t3/2.
(c)
height=104src=”./xi%20work,%20power%20and%20energy_files/image004.gif”alt=”5a1.gif (1325 bytes)” v:shapes=”_x0000_i1046″>

(d)
height=102src=”./xi%20work,%20power%20and%20energy_files/image005.gif”alt=”5a1i.gif (1265 bytes)” v:shapes=”_x0000_i1047″>

Q2. A simple pendulum of mass m and lengthl swings back and forth up to a maximum angle q0 with the vertical. When at an angle q, what is its (a) potential energy, (b) kinetic energy, (c) speed, and(d) tension?

Ans2.
height=149src=”./xi%20work,%20power%20and%20energy_files/image006.gif”alt=”5a2.gif (2041 bytes)” v:shapes=”_x0000_i1048″>
Taking the reference level at the lowest point R, we have
hP = l – l cos q0 = l(1 – cos q0)
hQ = l – l cos q = l(1 – cos q)
So (a) potential energy at Q relative to R will be
PE = mghQ
PE = mgl(1 – cos
q) ………(a)
(b) PE at P = mghP = mgl(1 – cos q0)
KE at P = 1/2 x mv2 = 0
so, total mechanical energy at P = mgl(1 – cos
q0) …….(i)
Now, if KQ is the KE at Q,
then using eq. (i)
mechanical energy at Q = KQ + mgl(1 – cos
q) ……….(ii)
But by conservation of mechanical energy between P and Q
KQ + mgl(1 – cos
q) = mgl(1 -cos q0)
i.e., KQ = mgl(cos q– cos q0) ……..(b)
(c) If v is the speed at point Q, from eq. (b)
1/2 x mv2 = mgl(cos q – cos q0)
i.e., v = height=17src=”./xi%20work,%20power%20and%20energy_files/image007.gif”alt=”i2.gif (1049 bytes)” v:shapes=”_x0000_i1049″>.
(d) If ‘E’ is the energy at
Ðq, then itis equal to mgl(1 – Cosq) + (1/2)mv2.
Since the energy remains constant throughout, E = Eo.
mgl(1 – Cosq) + (1/2)mv2 = mgl(1 -Cosqo)
or mv2 = 2mgl(Cosq – Cosqo)
Therefore, tension ‘T’ at q would begiven by
T = mv2/l + mgCos
q = mg Cosq + 2mg(Cosq– Cosqo)
or T = 3mgCosq – 2 mgCosqo.

Q3.What do youmean by work in the language of physics? Give its absolute and gravitationalunits. Give two illustrations of zero work, negative work and positive work.

Ans.(Try yourself).

Q4.How will youfind work done by a variable force mathematically and graphically?

Ans.(Try yourself).

Q5.What do youmean by conservative and non-conservative forces? Give their importantproperties.

Ans.(Try yourself).

Q6.What do youmean by gravitational potential energy? Show that gravitational potentialenergy is independent of the path followed.

Ans.(Try yourself).

Q7.If a body iskept on the top of a rough inclined plane, find the expression for
(i) work done in bringing it down to the bottom of the plane with constantvelocity
(ii) work done in moving it up the plane with constant acceleration
(iii) work done in moving it down the plane with constant acceleration.

Ans.(Try yourself).

## Conservation of energy Problem

A ball is dropped from a height of 10m. If the kinetic energy of the ball reduces by 40% after striking the ground, how high can the ball bounce back? (Elsy James asked)

## Motion under gravity

Two stones of mass 5 kg and 10 kg are dropped from the same height above the ground level. Which one will hit the ground first? Why?

Answer: Both the stones will reach the ground simultaneously (provided we neglect the air resistance (the viscosity) and buoyant force.

This is because acceleration due to gravity is independent of mass of the freely falling body.

## A few numerical problems from laws of motion

1. A worker pulls a 200. N packing crate with an applied force of 55.0 N.  The crate accelerates at a rate of 0.250  m/s(squared).  What is the coefficient of kinetic friction between the crate and the factory floor?
2. A student pulls a 150. N sled up a 28 degree slope at a constant speed by applying a force of 100. N.  Near the top of the hill he releases the sled.  With what acceleration does the sled go down the hill?
3. A 200. N crate rests on a ramp; the maximum angle just before it slips is 25 degrees with the horizontal.  What is the coefficient of static friction between the crate and the surface of the ramp?
4. A man pulls a sled with a weight of 200. N with a constant velocity across a horizontal snow surface.  If a force of 80N is being applied to the sled rope at an angle of 53 degrees to the ground, what is the coefficient of friction between the sled and the ground?
5. A jet plane is flying with  a constant speed along a straight line at an angle of 30degree  above the horizontal. The weight of plane is 86 500N. Its engine provides a forward thrust T of 103 000N. The lift of force L(directed perpendicular to the wings)and the force R of air resistance (directed opposite to the motion)act on the plane. Find L & R

## An interesting numerical problem from kinematics

George’s favorite food was bananas. One day he was walking with his friend, the man in the yellow hat, as he was eating a banana. They were in New York City and they were walking up the Empire State Building. Finally they made it to the top! George spied a nice young woman walking down 5th avenue balancing this hat on her head.  She was very noticeable for she was six feet tall! George, being the troublemaker he was, wanted to see if he could drop his banana peel in her hat. He noticed that she was jogging at a constant pace of 6 miles per hour. George was on the 102nd floor observatory, which is 1,224 feet from the ground.*
Assuming that George lands the banana peel in the hat……..
A)      How long will the banana peel be in the air? (Assuming there is no wind and the banana will drop straight down from the empire state building)
B)      What will the velocity of the banana peel be right before it lands in the woman’s hat?
C)      How far away must the woman’s hat be from the landing point of the banana peel (horizontally) when George makes his scandalous move and drops the peel.
D)      If George decides to throw another peel one second later to land it on the ground and trip the jogger, at what velocity must he throw it at so that it hits the ground at the same the original peel lands in the hat.

## Kinematics Numerical

1. A particle travels 20 m in 7th second and 24m in 9th sec. find initial velocity?
2. in a projectile motion , a body thrown from the ground ,at what angle both the vertical height and range will be equal?
3. The velocity v(cm/s) of a particle is given in terms of time t(in seconds) by the equation v = at+b/t+c. Dimension of a,b, and c are ?

## Horizontal projectile and freely falling body

A rifle at a height H aimed horizontally fires a bullet to the ground. At the same time , a bullet with the same mass in dropped from the same height. Neglecting air resistance, which one hits the ground first?Explain.

ED posted

Both will hit the ground simultaneously.

When a body is projected horizontally, its initial vertical velocity is zero and vertical acceleration is g, the acceleration due to gravity.

The values of a velocity and acceleration of a freely falling body are also the same.

So, both will hit the ground simultaneously

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