Mechanics Exercise Class Ⅳ

Review 1 Simple Harmonic Motion Velocity Acceleration 2 Energy 3 Pendulums Simple pendulum Physical pendulum

Review 4 Damped Harmonic Motion 5 Forced Oscillations and Resonance The greatest 6 Transverse and Longitudinal Waves Sinusoidal Waves

Review 7 Wave Speed on Stretched String The average power 8 Interference of Waves resonance 9 Standing Waves

Review 10 Sound Waves 11 Interference (m=0.1.2…) 12 The Doppler Effect Constructive interference Destructive interference (m=0.1.2…) 12 The Doppler Effect The speed of the detector The speed of the source

Review 13 Flow of Ideal Fluids =a constant =a constant 14 Bernoulli’s Equation =a constant

The block –springs system forms a linear simple harmonic oscillator, 1 Suppose that the two springs in figure have different spring constants k1 and k2. Show that the frequency f of oscillation of the block is then given by where f1 and f2 are the frequencies at which the block would oscillate if connected only to spring 1 or only to spring 2. Solution Key idea: The block –springs system forms a linear simple harmonic oscillator, with the block undergoing SHM. o x Assuming there is a small displacement x , then the spring 1 is Stretched for x and the spring 2 is compressed for x at the same time. From the Hook’s law we can write The coefficient of a simple spring

Using the Newton’s Second Low , we can obtain Thus the angular frequency is And the frequency f of oscillation of the block is

So the x-component of the force that the spring exerted on the mass is 2 A spring with spring constant k is attached to a mass m that is confined to move along a frictionless rail oriented perpendicular to the axis of the spring as indicated in the figure. The spring is initially unstretched and of length l0 when the mass is at the position x = 0 m in the indicated coordinate system. Show that when the mass is released from the point x along the rail, the oscillations occur but their oscillations are not simple harmonic oscillations. Solution: The mass is pulled out a distance x along the rail, the new total length of the spring is So the x-component of the force that the spring exerted on the mass is

From the diagram , so the x-component of the force is For the x-component of the force is Thus when the mass is released from the point x along the rail, the oscillations occur but their oscillations are not simple harmonic oscillations.

3 The figure gives the position of a 20 g block oscillating in SHM on the end of a spring. What are (a) the maximum kinetic energy of the block and (b) the number of times per second that maximum is reached? Solution:The key idea is that it’s a SHM. (a) From the figure, we can get the period T and amplitude of the system, they are so the angular frequency is The total mechanical energy of the SHM is

When the block is at its equilibrium point, it has a maximum kinetic energy, and it equals the total mechanical energy, that is (b) Because the frequency is the number of times per second that maximum is 50

4 Two sinusoidal waves , identical except for phase, travel in the same direction along a string and interfere to produce a resultant wave given by with x in meters and t in seconds. What are (a) the wavelength of the two waves, (b) the phase difference between them , and (c) their amplitude ? solution From the equation , we can get the wavelength of the two waves: (b) From the equation we can obtain the phase difference (c) Because we know , their amplitude is

5 A bat is flitting about in a cave , navigating via ultrasonic bleeps 5 A bat is flitting about in a cave , navigating via ultrasonic bleeps. Assume that the sound emission frequency of the bat is 39000Hz. During one fast swoop directly toward a flat wall surface , the bat is moving at 0.025 times the speed of sound in air. What frequency does the bat hear reflected off the wall? Solution: There are two Doppler shifts in this situation. First, the emitted wave strikes the wall, so the sound wave of frequency is (source moving toward the stationary wall)

(observer moving toward the Second , the wall reflects the wave of frequency and reflects it , so the frequency detected , will be given : (observer moving toward the stationary source)

6 A device to study the suction of flowing fluid is shown in the figure. When the fluid passing through the horizontal pipe the liquid in tank A can be drawn upward by the stream in the pipe. Assume that the upper tank is big enough so that the surface of the water in it does not descend obviously when the stream flows continuously. The height difference between the water surface in the upper tank and the pipe is h. The cross-sectional area of the pipe at the narrow place and the open end are S1 and S2, respectively. The density of the ideal fluid is ρ . Show that Solution: Choose the height of the pipe as the reference level for measuring elevations and point O at the surface of the water in the upper tank, point 1 at the narrow place and point 2 at the open end of the pipe.

Applying Bernoulli’s equation to a streamline passing through point O and 1: Then (1) As Applying Bernoulli’s equation to a streamline passing through point O and 2, we have: (2) So Apply the equation of continuity for point 1 and 2,we get (3)

Thus Substituting the above equation into Eq.(1) and noticing that S1<S2, then