Edexcel A-Level Physics: Oscillations and Nuclear Radiation

A mechanical system, such as a child's swing or a mass hanging on a spring, can be made to oscillate either freely or by a periodic driving force.

Explain the difference between free and forced oscillations, and explain what is meant by resonance. Go on to describe, with a real example, how increasing the damping of a system changes both the height and the width of its resonance peak.

(6 marks)

A mass is attached to the lower end of a vertical spring and set oscillating with simple harmonic motion. The amplitude of the oscillation is small enough that the spring obeys Hooke's law throughout.

Quantity	Value
Mass on spring, $m$m	0.250 kg
Spring constant, $k$k	40.0 N m⁻¹
Amplitude of oscillation, $A$A	0.080 m

(a) Calculate the period of the oscillation. (2 marks)

(b) Calculate the maximum speed of the mass during the oscillation. (2 marks)

(c) Calculate the maximum acceleration of the mass and the total energy of the oscillation. (2 marks)

A small object oscillates with simple harmonic motion. A motion sensor records its displacement $x$x from the equilibrium position at regular time intervals over one complete oscillation:

| Time $t$t / s | 0 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | |---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:| | Displacement $x$x / cm | +6.0 | +4.2 | 0 | −4.2 | −6.0 | −4.2 | 0 | +4.2 | +6.0 |

(a) Use the data to state the amplitude and the period of the oscillation, and hence calculate its frequency. (3 marks)

(b) Calculate the maximum speed of the object. (2 marks)

A sealed gamma point source is investigated with a Geiger-Müller (GM) tube connected to a ratemeter. With no source present, the tube records a steady background count rate of 30 counts per minute. The source is then placed so that its centre is 0.10 m from the GM tube window, and the ratemeter reads 750 counts per minute. The gamma radiation obeys an inverse-square law with distance, and absorption by the air is negligible.

(a) Calculate the corrected count rate due to the source alone at 0.10 m. (2 marks)

(b) The source is moved so that it is now 0.30 m from the GM tube window. Use the inverse-square law to calculate the corrected count rate the source would now produce. (2 marks)

(c) Hence state the count rate the ratemeter would actually display at 0.30 m. (1 mark)

A floating marker buoy of mass 36 kg bobs up and down on calm water. When the buoy is pushed down a small distance and released, the extra upthrust acts as a restoring force that is directly proportional to the depth it is pushed below its floating position. For this buoy the restoring force is 4000 N for every 1 m of vertical displacement, so it behaves like a mass on a spring of stiffness $k = 4000 \ \text{N m}^{-1}$k=4000 N m−1.

(a) Explain why the buoy oscillates with simple harmonic motion. (2 marks)

(b) Calculate the period and the frequency of the buoy's vertical oscillation. (2 marks)

An object moving with simple harmonic motion (SHM) has an acceleration $a$a that is related to its displacement $x$x from the equilibrium position by the equation $a = -\omega^2 x$a=−ω2x.

(a) State the defining condition for simple harmonic motion in words. (2 marks)

(b) Explain the significance of the negative sign in the equation $a = -\omega^2 x$a=−ω2x. (1 mark)