Problems Week 2 - Problems WeekZ 2~1 A pendulum on Earth(g...

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Unformatted text preview: Problems: WeekZ 2~1. A pendulum on Earth (g E : 9.8m/ 52) has a period of 1 see. What is its length? If you $3 take it to the moon what will its period be there if g M: ? If you want the period to be 1 sec on the moon by what factor must you change the length? 2—2. A spring—mass system is oscillating on a horizontal frictionless surface and its position‘is given by x: 0.05 m cos wt. For what values of x will (i) its kinetic energy be maximum, (ii) its potential energy be maximum (iii) its kinetic energy equal its potential energy, 2-3. If the system of problem 8—2 is hung from a ceiling, what will be the change in its ‘ frequency? Why? 2-5. 2-6. A wave on a stretched string is represented by y = 0.01 sin(6.28.x ~ 12.560)? —> ‘ where distances are in meters and times in seconds. (i) Is this longitudinal or transverse? (ii) What is its wavelength, frequency, velocity? A sinusoidal wave of amplitude A and frequency a) , travelling on a stretched string carries P=1A2w2£,where v: I 2 v ,u Joules of energy per second. Here T is the tension and v, the wave speed. Changing only one factor at a time how would P change if you (i) double A (ii) halve a) or (iii) increase Tby a factor of 3. A sine wave y, = A, sin(kx + cut) is launched on one string with velocityy = :Egi. At c x z 0, it encounters a second string where velocity is v' and gives rise to a reflected wave y,. = A sinUrx — cut) and a transmitted wave y, f: A, sin(k'x + w't) . (i) What is the . .. . . A. ~ relatlon between a)’ anda) ? Why? (11) What determines k'? (111) leen that ~4— = v v i v+v‘ and i : 2v A | show that during reflection there is a phase change of H if v' << v. , v + v 2—7. Using the result of problem 2-6 and the superposition principle, show that where the A incident and reflected waves combine, nodes appear at x = 0,11301. =1,2,...) and antinodes occur at x = (2n +1)—:1(n = 1,2,...) . 2—8. A wire of length 1m and mass 0.001kg has a tension of MN 111 it and is fixed at both ends. Calculate the frequencies of the 3rd and 5th harmonic modes. 2-9. In the standing wave experiment that you have performed (a) why does the lowest (11- — 1) mode require the largest tension (hanging mass)? (b) by what factor must you change the mass to get the 11 — 3 mode? , ,, 2-10., The intensity of a periodic sound wave of amplitude s," and frequency a) in a gas at pressure Po is ‘ P I=lsmzw2 y 0 2 v S C . . . Where y = E} and vs is the speed of sound. Calculate Sm for a1r1fP0=105N/m2, y = 1.4, v3: 330m/s and I: 10: lO'leatt/ m2; the quietest sound that can be heard. 2—11. A periodic sound wave can be thought of as a displacement wave s = Sm sin(k;x — cut) or a pressure wave P=PD —- Ps cos(kx—a)t) Why is the pressure ane always % out of phase with the displacement, that is, why is pressure variation maximum where displacement is zero and vice versa? 2-12. In air the speed of sound at room temperature is about 330m/s. What must be the wavelengths of mechanical waves to be called sound? 2—13; The speed of sound in a gas is written as C K. :JflfBT Where 7:4 Lm Cy .Why is there a “ 2/ ” in this equation? 2—14. Using the formula of 2—13 compare the speeds of sound in Helium [7 = 3m = 477119] and air [y = %,m ; 30mp]. 2-1 5. Compare the speed of sound in He (y = g) with the r.m.s. speed v atoms. "116' :51ch m of He ...
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