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How to solve questions on oscillation for Cambridge A level

cambridge oct/nov 2006 p4

Q1

Two vertical springs, each having spring constant k, support a mass. The lower spring is
attached to an oscillator as shown below

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The oscillator is switched off. The mass is displaced vertically and then released so that it
vibrates. During these vibrations, the springs are always extended. The vertical acceleration
a of the mass m is given by the expression
ma = –2kx,
where x is the vertical displacement of the mass from its equilibrium position.
Show that, for a mass of 240 g and springs with spring constant 3.0Ncm–1, the
frequency of vibration of the mass is approximately 8Hz

Solution

mass = 240g = 0.24kg

k = 3.0 Ncm-1 = 300 Nm-1

ma = –2kx

capture1……….i

0.24*a = -2* 300* x

a = -600x / 0.24

a = 2500x

substitute for a in equ i

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find the square root of both sides

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f = 8 Hz

The question is from cambridge may/june 2014 p41

A student investigates the energy changes of a mass oscillating on a vertical spring, as shown below

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The student draws a graph of the variation with displacement x of energy E of the oscillation, as shown below

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The student repeats the investigation but with a smaller amplitude. The maximum value of E
is now found to be 1.8 mJ.
Use graph above to determine the change in the amplitude

Solution

From the graph the maximum kinetic energy = 2.4 mJ

Change in Kinetic energy = 2.4 – 1.8 = 0.6 mJ

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Amplitude = 1.5 cm = 0.015 m

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When the amplitude change maximum energy E is 1.8m J

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Change in Amplitude = 1.5 cm – 1.3 cm

change in amplitude = 0.2 cm

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