KEY SOLUTION. 05/07/01 PHYSICS 223 Exam #1 NAME M 1 M 1. Fig. 1a Fig. 1b Fig. 1c

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1 KEY SOLUTION 05/07/01 PHYSICS 223 Exam #1 NAME Use g = 10 m/s 2 in your calculations. Wherever appropriate answers must include units. 1. Fig. 1a shows a spring, 20 cm long. The spring gets compressed 2 cm when a block of mass M 1 = 0.2 Kg. is attached to it; the mass M 1 is at rest as shown in Fig. 1b. Later on, another mass M 2 = 0.2 Kg is released (with zero speed) from above and undergoes an inelastic collision with M 1. See Fig.1c. The masses M 1 and M 2 remain well attached to each other (you can assume there exist glue in the surfaces of the masses.) There is not friction and the system is free to move along the vertical direction.) H M 2 g M 1 M 1 Fig. 1a Fig. 1b Fig. 1c 1A. The value of the spring constant "k", expressed in N/m, is: a) 100 b) 0.1 N/m c) 1 d) 10 N/m e) NA 1B. Consider Fig. 1c. Which of the following expressions is correct? a) The higher the height H (from which the mass M 2 is released) the higher the angular frequency of the SHM. b) Since the collision between M 1 and M 2 is inelastic, the whole energy carried by the mass M 2 before the collision is consumed by the interaction. Thus the whole system (M1+M2) remains at rest. c) The amplitude of oscillation increases as the value of H increases, which causes the period of oscillations to increase. d) The angular frequency of the SHM does not depend on the value of M 2. e) NA 2. The value of H in the previous problem is 5 cm (See Fig. 1c). 2A. The angular frequency "ωο" of the oscillations (in units of rad/sec) is (round your answer to the closest integer number): a) 22 b) 250 c) 3 d) 16 e) NA

2 2B. The amplitude of the SHM motion is (in cm): a) 3.7 b) 0 c) 2.2 d) 3.0 e) NA 3. 3A You and your classmates wish to tune the note A 3 on a piano to its proper frequency of 220 Hz. You use a tuning fork whose frequency has a very precise value of 220 Hz, and observe a beat frequency of 6 Hz. Which one of your classmates conclusion is 100% correct: a) The frequency of the string is definitely 226 Hz. b) Change the tuning fork for one that establishes a zero beat frequency with the piano. c) This problem is too difficult to solve, you better get ride of the piano. d) If you tighten the piano string the beat frequency will definitely increase. e) All the above statements are inaccurate. 3B When a guitar string is plucked (indicate which of the following statement is correct) a) The wavelength of the wave it produces in air is always the same as the wavelength on the string. b) The frequency of the wave on the string is definitely higher than the frequency of the wave in the air. c) The resonance frequencies associated with that particular guitar string depend on the type of song you are playing. d) The velocity of the wave in the string is the same as the velocity of the wave in air. e) All the above statements are wrong. 4. The figure below shows a 0.5 meter long string (with one end attached to the wall) of linear mass density 100 g/m, under a tension of 1.6 N. At one end a person shakes the string up and down with an amplitude of 4 cm. Physics 223 student L 4A 4B The speed of the reflected waves traveling along the string is: a) 4 m/s b) 0.12 m/s c) 5 m/s d) 2 m/s e) NA The person shakes the string with either 5 Hz or 6 hertz (one case at a time). Indicate which expression is correct: a) Both cases satisfy the conditions to establish resonant standing waves in the string b) Since the reflected waves travel at a different speed than the incident wave, no resonance pattern is possible to establish along the string. c) Since only one end of the string is fixed, no resonant standing wave can be set in the string.

3 d) Only one case satisfies the conditions for a resonant standing wave e) NA 5. 5A. Two connected wires with linear mass densities that are related by µ 1 = 4 µ 2 are under some tension. (Assume the wires are very long so we can omit the effects of the reflected wave from the wall) 1 2 Figure for question 5A only A wave of frequency 120 Hz and wavelength λ =10 cm is set on the left wire a) The wavelength on the right side will be 40 cm. b) The velocity of the wave will be the same in both sides c) While the wave propagates to the right side the different components of the string shake horizontally back and forth around their equilibrium positions. d) The tension on the wire at the left side is twice the tension on the left side. e) NA 5B. Write an equation describing a sinusoidal transverse wave traveling on a cord in the ( X) direction with a wavelength of 10 cm, a frequency of 120 Hz and an amplitude of 2 cm. Assume the cord has a uniform linear mass density. Write clearly and include the proper units. ANSWER 2π 120 y( x, t) = 0.02m SIN x + 2π t 0.1m sec 6. 6A The figure indicates the direction of motion of a sound source and a detector for 4 situations in stationary air. In all these cases the detector is at the right side of the source, and the size of the arrows (which indicate the speeds of the source and observer) are the same.

4 Source detector ( f ) ( f ) 0 speed f < f f = f f > f f < f f = f f > f 0 speed f < f f = f f > f f < f f = f f > f 6B. The 14,000 Hz whine of the turbines in the jet engines of an aircraft moving with speed 210 m/s is heard at what frequency by the pilot of a second craft trying to overtake the first at a speed of 260 m/s? (Assume the speed of the sound is 340 m/s) a) 260 Hz b) Hz c) Hz d) Hz e) NA 7. The electric field of a certain electromagnetic field is given by E x = 0, E y = 0, E z = 48 V/m COS (10 7 m -1 x - ω t) Y E Z B X 7A. In the diagram above draw the electric field and the magnetic field vectors at the position x=0 and at the instant t=0. In addition, write down below the expressions for the component of the magnetic field of this wave are: B m =E m /c B x = 0 B z = 0 B y = - 1.6x10-7 T COS (10 7 m -1 x - ω t)

5 7B. Consider the same wave given in problem 7A. The wavelength λ and the direction of propagation are: a) 10 7 m, (-X axis ) b) 628 nm, ( +X axis ) c) 10-7 m, ( X axis) d) 209 nm, (+X axis) e) NA 8. 8A The figure below shows 3 electromagnetic waves (incident, reflected and transmitted) at an air-glass interface. Only the orientation of the electric fields (at a given time) are indicated. In the circles provided in the graph use "." dots or "x" crosses to indicate the corresponding magnetic fields orientation of the electromagnetic waves. Air c E transm X v E inc X. E reflec Glass v 8B In the figure below shows 2 charges initially at rest and a traveling electromagnetic wave (the wave starts its propagation at point H.) Which one of the two charges will be excited by the electromagnetic wave? ANSWER +q Draw an arrow to indicate the direction of the linear momentum carried by the electromagnetic wave. +q B - q E H

6 9. The figure below shows a frontal view of a thin ring, a solid sphere and a solid cylinder which are undergoing SHM (simple harmonic motion.) All of them have the same radius R=25 cm, but different mass. (the length of the cylinder is L= 0.5 R). The axis of rotation is perpendicular to this page. Support M 0 2 M 0 4 M 0 Ring Sphere Cylinder 9A The number of oscillations that the sphere undergoes per second is (approximately) a) 1.6 b) 3.2 c) 0.17 d) 1 e) NA 9B Arrange the solid pendulums according to their period of oscillations. Greatest first: ANSWER RING, CYLINDER, SPHERE 10. A damped oscillator has mass m= 500 g, k = 0.75 N/cm. The damping constant is b= 0.05 Kg/sec. 10A Find the period "T" and the time t=t 1 it takes for the amplitude of the damped oscillations to drop to one third of its initial value. How many oscillation does the damped oscillator make from t=0 to t=t 1. ANSWER t = 0.5 s t 1 = 22 s 43 oscillations 10B Find the time t=t 2 it takes for the mechanical energy of the damped oscillations to drop to one third its initial value. How many oscillation does the damped oscillator make from t=0 to t=t 2. ANSWER t 2 = 11 s 21 oscillations

7 11 In a horizontal table a mass of 0.5 Kg is attached to a spring ( k = 50 N/m). The mass is stretched 5 cm from its equilibrium position, released with zero speed and undergoes SHM. 11A Indicate which of the following function(s) describes correctly the position of the mass. 1 x ( t) = 5cm COS( 10s t + / 9 ) ANSWER: Yes No x 1 π 1 2 t I will accept both answers because the problem did not say explicitly that t=0 at the time of release ( t) = 5cm COS(20π s ) ANSWER: Yes No. 1 x ( t) = 10cm COS(20π s ) ANSWER: Yes No. 2 11B Indicate which of the following functions describes correctly the velocity of the mass. x t cm 1( ) = 50 COS( 10s 1 t ) ANSWER: Yes No x s cm 1 2 t s For the same reason stated above ( t) = 100π COS(20π s ) ANSWER: Yes No. FORMULAS THAT MAY HELP 2 d x b d x k Damped simple harmonic oscillator + + x = 0 2 t m t m αt x( t) = Ae COS( ωt + φ) where b 2 2 ( b ) 2 α = ω = ω 2m 0 2m Waves traveling along a string: v = (T/ µ) 1/2 ω = 2π f

8 R I = MR 2 I = 2/5 MR 2 I = 1/2 MR 2 Ring Sphere Cylinder ' v sound ± vobserver f = f where f is the frequency emitted by the source v ± v sound source POYNTING VECTOR S = 1 ExB µ 0 E y Bz Em = Bm = x t c

08/26/09 PHYSICS 223 Exam-2 NAME Please write down your name also on the back side of this exam

08/26/09 PHYSICS 223 Exam-2 NAME Please write down your name also on the back side of this exam 08/6/09 PHYSICS 3 Exam- NAME Please write down your name also on the back side of this exam 1. The figure shows a container-a holding an ideal gas at pressure 3.0 x 10 5 N/m and a temperature of 300K.