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A position sensor tracks a component undergoing simple harmonic motion. Its displacement from equilibrium is modelled by
where the phase angle is in radians. The amplitude is the period is and the initial phase angle is
State the phase angle at
Calculate the phase angle at in radians.
Calculate the displacement of the component at
State the increase in phase angle during one complete oscillation, in radians.
An oscillator has a period of Four proposed graphs show its acceleration against displacement measured from equilibrium. Which trace is consistent with the oscillator undergoing simple harmonic motion?
Answer: A. For simple harmonic motion, the graph of against is a straight line through the origin with gradient From so the required gradient is Trace P has this negative gradient and passes through the origin.
B: uses a negative gradient through the origin, but its magnitude is about confusing with
C: has the required gradient but does not pass through the origin, so acceleration is not proportional to displacement.
D: has a positive gradient, so acceleration is in the direction of displacement rather than towards equilibrium.
Remember: SHM requires acceleration to be proportional to displacement and opposite in direction.
A cord and a spring each carry a sinusoidal travelling disturbance along the equilibrium -axis. Figure 1 shows the displacement of particles from equilibrium at and plotted against each particle’s equilibrium coordinate In the upper panel, is the cord-particle displacement perpendicular to with positive upwards. In the lower panel, is the spring-particle displacement along with positive in the direction. Each disturbance retains its profile as it travels.
Determine the direction of propagation and the speed of the disturbance on the cord.
State the wavelength of the disturbance on the cord.
At determine the instantaneous velocity of the cord particle at Give your answer with its direction.
The spring disturbance is now launched in the opposite direction, with its speed and wavelength unchanged. Describe how its profile at is displaced relative to its profile at State whether the disturbance is still longitudinal, and explain your answer.
In a laboratory, a laser illuminates both slits of a double-slit plate at normal incidence. The separation of adjacent bright fringes is measured on a screen parallel to the plate. A reference laser has wavelength With the original plate and a screen distance of the measured fringe separation is
Show that the slit separation of the original plate is approximately
The screen is moved to a distance of with the original plate and reference laser unchanged. Determine the fringe separation.
A replacement plate is used with the reference laser and the screen at The measured fringe separation is Determine the slit separation of the replacement plate.
The replacement plate is used with an unknown laser wavelength. The screen is at and the fringe separation is Determine the wavelength of this laser.
In a further run, an unknown laser gives a fringe separation of with the screen at but the identity of the slit plate used was not recorded. A technician claims that the laser wavelength is the same as that of the reference laser. Determine whether this claim is justified, supporting your answer quantitatively.
A string on a musical instrument is fixed at both ends and vibrates in the mode shown in Figure 1.
State the number of displacement nodes and the number of antinodes shown. Count the fixed ends as nodes.
The string has length and the wave speed on it is For the mode shown, calculate the wavelength and frequency. Give both answers to 3 significant figures.
Compare the oscillation amplitudes at A and B, and state their phase difference.
Sound travels at in a pipe of length The pipe is closed at one end and open at the other. Calculate the wavelength and frequency of the third harmonic. Ignore end correction.
A source emits light with wavelength 600 nm and moves directly away from an observer at Take the speed of light as What is the magnitude of the relative change in wavelength, expressed as a percentage?
Answer: C. For light, the magnitude of the relative wavelength change is approximately Thus
A: converts the fractional change to a percentage using a factor of 10 rather than 100.
B: introduces an erroneous factor of
D: shifts the decimal point one place too far when converting the fraction to a percentage.
Remember: multiply a fractional change by 100 to express it as a percentage.
A sound emitter moves uniformly through still air. At one instant, the distance between adjacent wavefronts is 0.646 m ahead of the emitter and 0.714 m behind it. The speed of sound in the air is 340 m s⁻¹. In a second arrangement, the emitter is stationary and a detector moves towards it at the speed found in (a). The emitter has the same frequency in both arrangements.
Determine the emitted frequency and the speed of the emitter. Show your reasoning.
Sketch wavefront diagrams for (i) the sound emitter moving towards a stationary detector and (ii) a stationary emitter with the detector moving towards it. Indicate the directions of motion and the relative spacing of the wavefronts.
Calculate the frequency detected in each sound arrangement. Give your answers to three significant figures.
A light beacon moves towards an observer. State how the detected frequency and wavelength compare with those emitted, and state one relevant difference between light waves and sound waves.
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