Physics 106 Group Problems Summer 2015 Oscillations and Waves

Transcription

1 Physics 106 Group Problems Summer 2015 Oscillations and Waves Name: 1. (5 points) The tension in a string with a linear mass density of kg/m is 0.40 N. What is the frequency of a sinusoidal wave with a wavelength λ = 0.2 m on this string? F λf = µ f = 100 Hz 2. (5 points) A 3 kg block, attached to a spring, executes simple harmonic motion according to 2 cos(50t) where x is in meters and t is in seconds. What is the spring constant? From the harmonic equation of motion x(t) = A cos(ωt) we see that ω = 50 k ω = m k = 7500 N/m TA: 1

2 3. (10 points) Consider a block of mass m = 5 kg that is hanging on a light spring as shown The natural length of the spring is l 0 = 10 cm, and the spring constant is k = 20 N/cm. At t = 0, the block is released from rest at the position where the spring is l = 20 cm in length, then it begins a simple harmonic oscillation. (a) What is the length l eq of the spring when the block reaches the equilibrium position (F net = 0)? The gravitational force from the block with mass m equals the stretching force on the spring. mg = k(l eq l 0 ) l eq = l 0 + mg 5 10 = 10 + k 20 l eq = 12.5 cm (b) Find the period T, angular frequency ω, amplitude A of the oscillation. Find the maximum speed and acceleration of the block. m 5 T = 2π k = 2π 2000 = s ω = 2π = 20 s 1 T A = l l eq = 7.5 cm v max = Aω = = 1.5 m/s a max = Aω 2 = = 30 m/s 2 (c) Assuming the gravitational potential energy is 0 at x = 0 (the base of the spring ), calculate the total energy of the system (block and spring). K i = 0 U i = mgl k(l l 0) 2 = = 0 E = K i + U i = 0 2

3 4. (10 points) Two train whistles, A and A, each have a frequency of 392 Hz. A is stationary and A is moving toward the right at speed of 35.0 m/s. A listener is between the two whistles and is moving toward the right with the speed of 15.0 m/s. No wind is blowing. Assume that the speed of sound is 343 m/s. (a) What is the frequency of whistle A as heard by the listener? Doppler Effect equation is v sound ± v obs f obs = f src v sound v src where we take the top sign when the motion is towards the other thing. We take the bottom sign when the motion is away from the other thing. For the frequency of whistle A heard by the observer f obs,a we have the observer moving away from the source. v sound v obs f obs,a = f src v sound + 0 f obs,a = Hz (b) What is the frequency of whistle B as heard by the listener? The source is moving away from the observer so we take the +v src is moving towards the source +v obs. The Doppler s equation becomes: v sound + v obs f obs,b = f src v sound + v src f obs,b = Hz (c) What is the beat frequency detected by the listener? and the observer The superposition of two sound waves with nearly identical frequencies are perceived by the listener as oscillations of the sound at beat frequency equal to the absolute difference of frequencies of the two waves. f beat = f obs,b f obs,a f beat = 3.6 Hz 3

4 5. (10 points) A girl is sitting near the open window of a train that is moving at a velocity of m/s to the east. The girls uncle stands near the tracks and watches the train move away. The locomotive whistle emits soundfrequency Hz. Use v sound = m/s as the speed of sound. (a) If the air is still, what frequency does the uncle hear? Doppler s equation for stationary observer gives us: ( f obs = f src 1 + v source v sound ) 1 f obs = Hz (b) A wind begins to blow from the east at m/s. What frequency does the uncle now hear? This time the velocity of the sound will increase due to the wind blowing in the same direction. v sound,b = v sound v wind = 353 m/s Plugging the new speed of sound in the equation from part a we get for the f obs f obs = Hz 4

5 6. (10 points) A block of mass m is attached to a massless spring with spring constant k, and is set oscillating over a frictionless horizontal surface as shown in Figure (a). Figure (b) shows the blocks kinetic energy versus its position x. At x = 5 cm, the blocks kinetic energy K s = 3 J. (a) What is the spring constant? From the diagram we can get the values of A = 0.1 m and for K max K max = (1 + 1/3)K s K max = 4 J K max = 1 2 ka2 k = 800 N/m (b) If the blocks maximum acceleration is a max = 40 m/s 2 what is the mass of the block? Equating the force from the deformation of the spring ka to the Newton s force ma max we get ma max = ka m = 2 kg 5

6 7. (10 points) The end point of a spring oscillates with a period of 2.0 s when a block with mass m is attached to it. When this mass is increased by 2.0 kg, the period is found to be 3.0 s. (a) What is the mass of the block? m T = 2π k T old m + 2 = T new m m = 1.6 kg (b) If the block oscillates with a period of 1.0 s and maximum speed of m/s. What is the maximum compression (in cm) of the spring? K = E spring 1 2 mv2 max = 1 2 kx2 max m T = 2π k x max = 5 cm 6