LODO Prueba
3.4 PERFORACIÓN DE LA SEGUNDA FASE 12 ¼”
whenever he thinks about his guru, the guru immediately knows about it. Calculate the minimum possible time interval between the yogi thinking about the guru and the guru knowing about it.
2. A suitcase kept on a shop's rack is measured 50 cm x 25 cm x 10 cm by the shop's owner. A traveller takes this suitcase in a train moving with velocity 06c. If the suitcase is placed with its length along the train's velocity, find the dimensions measured by (a) the traveller and (b) a ground observer.
3. The length of a rod is exactly 1 m when measured at rest. What will be its length when it moves at a speed of (a) 3 x 10 5 m/s, (b) 3 x 10 6 n/s and (c) 3 x 107 m/s ?
4. A person standing on a platform finds that a train moving with velocity 0'6c takes one second to pass by him. Find (a) the length of the train as seen by the person and (b) the rest length of the train.
5. An aeroplane travels over a rectangular field 100 m x 50 m, parallel to its length. What should be the speed of the plane so that the field becomes square in the plane frame ?
6. The rest distance between Patna and Delhi is 1000 km. A nonstop train travels at 360 km/h. (a) What is the distance between Patna and Delhi in the train frame ? (b) How much time elapses in the train frame between Patna and Delhi ?
7. A person travels by a car at a speed of 180 km/h. It takes exactly 10 hours by his wristwatch to go from the station A to the station B. (a) What is the rest distance between the two stations ? (b) How much time is taken in the road frame by the car to go from the station A to the station B ?
8. A person travels on a spaceship moving at a speed of 5q/13. (a) Find the time interval calculated by him between the consecutive birthday celebrations of his friend on the earth, (b) Find the time interval calculated by the friend on the earth between the consecutive birthday celebrations of the traveller.
9. According to the station clocks, two babies are born at the same instant, one in Howrah and other in Delhi, (a) Who is elder in the frame of 2301 Up Rajdhani Express going from Howrah to Delhi ? (b) Who is elder in the frame of 2302 Dn Rajdhani Express going from Delhi to Howrah.
10. Two babies are born in a moving train, one in the compartment adjacent to the engine and other in the compartment adjacent to the guard. According to the train frame, the babies are born at the same instant of time. Who is elder according to the ground frame ? 11. Suppose Swarglok (heaven) is in constant motion at a
speed of 0'9999c with respect to the earth. According to the earth's frame, how much time passes on the earth before one day passes on Swarglok ?
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12. If a person lives on the average 100 years in his rest frame, how long does he live in the earth frame if he spends all his life on a spaceship going at 60% of the speed of light.
13. An electric bulb, connected to a make and break power supply, switches off and on every second in its rest frame. What is the frequency of its switching off and on as seen from a spaceship travelling at a speed 0'8c ? 14. A person travelling by a car moving at 100 km/h finds
that his wristwatch agrees with the clock on a tower A. By what amount will his wristwatch lag or lead the clock on another tower B, 1000 km (in the earth's frame) from the tower A when the car reaches there ?
15. At what speed the volume of an object shrinks to half its rest value ?
16. A particular particle created in a nuclear reactor leaves a 1 cm track before decaying. Assuming that the particle moved at 0'995c, calculate the life of the particle (a) in the lab frame and (b) in the frame of the particle. 17. By what fraction does the mass of a spring change when
it is compressed by 1 cm ? The mass of the spring is 200 g at its natural length and the spring constant is 500 N/m.
18. Find the increase in mass when 1 kg of water is heated from 0°C to 100°C. Specific heat capacity of water = 4200 JAg-K.
19. Find the loss in the mass of 1 mole of an ideal monatomic gas kept in a rigid container as it cools down by 10°C. The gas constant R - 8"3 J/mol-K.
20. By what fraction does the mass of a boy increase when he starts running at a speed of 12 km/h ?
21. A 100 W bulb together with its power supply is suspended from a sensitive balance. Find the change in the mass recorded after the bulb remains on for 1 year. 22. The energy from the sun reaches just outside the earth's atmosphere at a rate of 1400 W/m2. The distance
between the sun and the earth is 1'5 x 1011 m.
(a) Calculate the rate at which the sun is losing its mass. (b) How long will the sun last assuming a constant decay at this rate ? The present mass of the sun is 2 x 10 30 kg.
23. An electron and a positron moving at small speeds collide and annihilate each other. Find the energy of the resulting gamma photon.
24. Find the mass, the kinetic energy and the momentum of an electron moving at 0'8e.
25. Through what potential difference should an electron be accelerated to give it a speed of (a) 0'6c, (b) 0'9c and (c) 0 99c ?
26. Find the speed of an electron with kinetic energy (a) 1 eV, (b) 10 keV and (c) 10 MeV.
27. What is the kinetic energy of an electron in electronvolts with mass equal to double its rest mass ?
28. Find the speed at which the kinetic energy of a particle will differ by 1% from its nonrelativistic value ^ m0 v
ANSWERS OBJECTIVE I 1. (d) 2. (c) 3. (c) 4. (b) 5. (b) 6. (b) 7. (b) OBJECTIVE II 1. (b), (d) 2. (c), (d) 3. (b) 4. (c) 5. (b), (c) 6. (a) EXERCISES 1. 1/300 s 2. (a) 50 cm x 25 cm x 10 cm (b) 40 cm x 25
c m
x 10 cm 3. (a) 0-9999995 m (b) 0 99995 m (c) 0 995 m 4. (a) 1-8 x 10 8 m (b) 2"25 x 10 8 m 5. 0'866c6. (a) 56 nm less than 1000 km 500 (b) 0"56 ns less than - g - min 7. (a) 25 nm more than 1800 km
(b) 0'5 ns more than 10 hours 13
8. — y in both cases
9. (a) Delhi baby is elder (b) Howrah baby is elder
10. the baby adjacent to the guard is elder 11. 70-7 days 12. 125 y 13. 0 6 s"1 14. will lag by 0'154 ns 16. (a) 33-5 ps (b) 3*35 ps 17. 1'4 x 10 "18 18. 4-7 x 10"12 kg 19. 1-38 x 10", 5kg 20. 6T7 x 10"17 21. 3-5 x 10 "8 kg 22. (a) 4-4 x 10 9 kgte (b) 1'44 x 10 13 y 23. 1'02 MeV 24. 15-2 x 10 "31 kg, 5-5 x 10 "14 J, 3*65 x 10 _22 kg-m/s 25. (a) 128 kV (b) 661 kV (c) 3'1MV 26. (a) 5-92 x 10 5 m/s (b) 5"85 x 10 7 m/s (c) 2"996 x 10 8 27. 511 keV 28. 3-46 x 107 m/s
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6. Test dimensionally if the equation v2 = u 2 + 2ax may be
correct.
Solution : There are three terms in this equation u \ u" and 2a.r. The equation may be correct if the dimensions of these three terms are equal.
Iv' - L2T~2 _ L2T" 2
[2ax] - [a] [x] L = L T Thus, the equation may be correct.
7. The distance covered by a particle in time t is given by x = a + bt + ct2 + dt3; find the dimensions of a, b, c and d.
Solution : The equation contains five terms. All of them should have the same dimensions. Since [x] = length, each of the remaining four must have the dimension of length.
Thus, [a] = length = L
[bt]=L, or, [b] = LT"1. [d 2] - L,
and [dt3] = L,
or, [c] = LT . or, [d] = LT "3
8. If the centripetal force is of the form m " v b r c, find the
values of a, b and c. Solution : Dimensionally,
Force = (Mass) a x (velocity) b x (length) c
or, MLT "2 = M a(Lb T "6) Lc = M Q L6 *c T " 6 Equating the exponents of similar quantities,
a = 1, b + c-1, - b = - 2
1 u o 1 J- m v '
or, a = 1, b •= 2, c - - 1 or, F = r
9. When a solid sphere moves through a liquid, the liquid opposes the motion with a force F. The magnitude of F depends on the coefficient of viscosity ri of the liquid, the radius r of the sphere and the speed v of the sphere.
Assuming that F is proportional to different powers of these quantities,guess a formula for F using the method of dimensions.
Solution : Suppose the formula is F=kr\arbvc
Then, MLT "2 = [ M L " ' T ^ L ' _M«L- o * b * eT- « - e
Equating the exponents of M, L and T from both sides, a = 1
- a + 6 + e = l - a - c = - 2.
Solving these, a = 1, b = 1, ana c = 1. Thus,the formula for F is F = kqrv.
10. The heat produced in a wire carrying an electric current depends on the current, the resistance and the time. Assuming that the dependence is of the product of powers type, guess an equation between these quantities using dimensional analysis. The dimensional formula of resistance is ML21 "2T 3 and heat is a form of energy. Solution : Let the heat produced be H, the current through
the wire be I, the resistance be R and the time be t. Since heat is a form of energy, its dimensional formula i s M L2T "2. •
Let us assume that the required equation is H-kl"Rbtc,
where fe is a dimensionless constant. Writing dimensions of both sides,
ML2T 2 = I "(ML2I ~2T 3) 6 T c
Equating the exponents,
6 = 1
26 = 2
- 36 + c = - 2 a - 2b = 0.
Solving these,we get a = 2, 6 = 1 and c = 1. Thus,the required equation is H = kl2 Rt.
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QUESTIONS F O R S H O R T A N S W E R
1. The metre is defined as the distance travelled by light in 299 799 45^ s e c o nd - Why didn't people choose some easier number such as 3QQ ^ second ? Why not 1 second ?
2. What are the dimensions of : (a) volume of a cube of edge a, (b) volume of a sphere of radius a,
(c) the ratio of the volume of a cube of edge a to the volume of a sphere of radius a ?
3. Suppose you are told that the linear size of everything in the universe has been doubled overnight. Can you test this statement by .measuring sizes with a metre stick ? Can you test it by using the fact that the speed of light is a universal constant and has not changed ? What will happen if all the clocks in the universe also start running at half the speed ?
•1. If all the terms in an equation have same units, is it necessary that they have same dimensions ? If all the terms in an equation have same dimensions, is it necessary that they have same units ?
5. If two quantities have same dimensions, do they represent same physical content ?
6. It is desirable that the standards of units be easily available, invariable, indestructible and easily reproducible. If we use foot of a person as a standard unit of length, which of the above features are present and which are not ?
7. Suggest a way to measure :
(a) the thickness of a sheet of paper,
(b) the distance between the sun and the moon.
O B J E C T I V E I
/ I . Which of the following sets cannot enter into the list of fundamental quantities in any system of units ? - (a) length, mass and velocity,
^ ^ l e n g t h , time and velocity, (c) mass, time and velocity,
length, time and mass.
2. A physical quantity is measured and the result is expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then
(a) ri oc size of u (b) n « u1
(c) /u
L, T and x,
(c) may be represented in terms of L, T and x if a = 0,
^Jjtf may be represented in terms of L, T and x if a * 0.
4. A dimensionless quantity
(a) never has a unit, (b) always has a unit, (c) may have a unit, (d) does not exist. 5. A unitless quantity
^Jtttf'never has a nonzero dimension, (b) always has a nonzero dimension, (c) may have a nonzero dimension, (d) does not exist.
Suppose a quantity ,-v can be dimensionally represented in terms of M, L and T, that is, [x] = M° Lb Tc\ The
quantity mass
(a) can always be dimensionally represented in terms of L, T and x,
(b) can never be dimensoinally represented in terms of
6.
J
dx 1J 'lax - x 2
The value of n is (a) 0 (b) - 1
(c) 1 (d) none of these.
You may use dimensional analysis to solve the problem.
O B J E C T I V E II
1. The dimensions ML"' T 2 may correspond to
(a) work done by a force (b) linear momentum ^e^pressure
v{<J) energy per unit volume. 2. Choose the correct statement(s) :
•^-fS) A dimensionally correct equation may be correct. vttT) A dimensionally correct equation may be incorrect. » (c) A dimensionally incorrect equation may be correct. A dimensionally incorrect equation may be incorrect.
3. Choose the correct statement(s) :
Jaf All quantities may be represented dimensionally in tepfns of the base quantities.
sfb) A base quantity cannot be represented dimensionally
iryterms of the rest of the base quantities.
Njc) The dimension of a base quantity in other base quantities is always zero.
(d) The dimension of a derived quantify is never zero in any base quantity.
E X E R C I S E S
1. Find the dimensions of (c) torque T and (d) moment of interia /. (a) linear momentum, (b) frequency and (c) pressure. Some of the equations involving these quantities are 2. Find the dimensions of
f = F.r and I 0 , - 0 , co., -co,
<0 = " > a =
/_, - t., - t,
The symbols have standard meanings. 3. Find the dimensions of
(a) electric field E. (b) magnetic field B (c) magnetic permeability p0.
The relevant equations are Ho'
and
F = qE, F = qvB, and B = 2 n a 4.
where F is force, q is charge, v is speed, I is current, and a is distance.
Find the dimensions of
(a) electric dipole moment p and (b) magnetic dipole moment M.
The defining equations are p = q.d and M = IA
where d is distance, A is area, q is charge and I is current.
5. Find the dimensions of Planck's constant h from the equation E = hv where E is the energy and v is the frequency.
6. Find the dimensions of (a) the specific heat capacity c,
(b) the coefficient of linear expansion a and tc) the gas constant R.
Some of the equations involving these quantities are Q = mc(T, - T,), I, = Z0[l + a(T.2 - T,)] and PV = nRT.
7. Taking force, length and time to be the fundamental quantities find the dimensions of
(a) density, (b) pressure, (c) momentum and (d) energy.
8. Suppose the acceleration due to gravity at a place is 10 m/s". Find its value in cm/(minute)".
9. The average speed of a snail is 0'020 miles/hour and that of a leopard is 70 miles/hour. Convert these speeds in SI units.
10. The height of mercury column in a barometer in a Calcutta laboratory was recorded to be 75 cm. Calculate this pressure in SI and CGS units using the following data : Specific gravity of mercury = 13'6, Density of water = 10 kg/m , g = 9 '8 m/s" at Calcutta. Pressure = hpg in usual symbols.
11. 12.
1 3 .
Express the power of a 100. watt bulb in CGS unit. The normal duration of I.Sc. Physics practical period in Indian colleges is 100 minutes. Express this period in :
microcenturies. 1 microcentury = 10 " x 100 years. How ; many microcenturies did you sleep yesterday ?
The surface tension of water is 72 dyne/cm. Convert it in SI unit.
14. The kinetic energy if of a rotating body depends on its moment of inertia I and its angular speed co. Assuming the relation to be K = klaco where k is a dimensionless
constant, find a and b. Moment of inertia of a sphere
2 2
about its diameter is -Mr . 5
15. Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.
16. Let I = current through a conductor, R = its resistance and V = potential difference across its ends. According to Ohm's law, product of two of these quantities equals the third. Obtain Ohm's law from dimensional analysis. Dimensional formulae for R and V are ML21~2T 3 and
ML2T -3!"1 respectively.
17. The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.