Maths › Further Mechanics 2 › Oscillations on strings and springs
Oscillations on strings and springs
Hooke's law supplies a restoring force proportional to displacement, which is exactly the simple harmonic condition. The only care needed is where the motion is centred and whether the string stays taut.
Builds on Simple harmonic motion and Elastic potential energy.
IN THIS TOPIC
- Prove that a mass on a spring moves with simple harmonic motion and find ω.
- Locate the centre of the oscillation at the equilibrium position.
- Recognise when a string goes slack and the motion stops being simple harmonic.
WHAT YOU PROBABLY THINK
A mass hanging on a spring oscillates about the position where the spring has its natural length.
Why Hooke's law gives oscillation
Measure x from the equilibrium position rather than from the natural length. At equilibrium the tension already balances the weight, so displacing the particle by x changes the tension by λx/l and nothing else. The equation of motion is therefore:
Gravity has dropped out entirely, which is why the oscillation is centred on the equilibrium position and not on the natural length: the opening claim confuses the two. The weight decides where the centre is; it has no effect at all on the period.
WORKED EXAMPLE
A mass on a spring
A particle of mass 0.5 kg hangs on a spring of natural length 0.4 m and modulus 20 N. Find the equilibrium extension and the period of small oscillations, taking g = 9.8 m/s².
Equilibrium: 20e/0.4 = 0.5(9.8), so 50e = 4.9 and e = 0.098 m.
ω² = λ/(ml) = 20/(0.5 × 0.4) = 100, so ω = 10 rad/s.
T = 2π/10 = 0.628 s, and the mass would have to quadruple to double it.
When the string goes slack
A spring pushes as well as pulls, so the motion is simple harmonic throughout however large the amplitude. A string cannot push, so as soon as the particle rises above the natural length the tension vanishes and only gravity acts: the particle moves freely until the string tightens again.
So compare the amplitude with the equilibrium extension. If the amplitude is smaller the string stays taut and the motion is simple harmonic all the way. If it is larger, the motion is simple harmonic in part and free under gravity in the rest, and the two pieces have to be handled separately and joined at the moment the string goes slack.
YOUR TURN
Released from the natural length
The same particle is pulled up to the point where the spring has its natural length and released from rest. Find its greatest speed, and say what would change if the spring were a string.
Show the working
The release point is 0.098 m above equilibrium and the particle starts at rest there, so the amplitude is 0.098 m.
Greatest speed = aω = 0.098 × 10 = 0.98 m/s, at the equilibrium position.
With a string, the particle would reach exactly the natural length at the top of each swing, where the tension is zero.
Any larger amplitude and the string would go slack there, and the particle would rise as a free body before the string snapped taut again.
THE EXAM BIT
- Measure the displacement from the equilibrium position and say that you are doing so.
- Show the weight cancelling; that step is what makes it simple harmonic and it carries marks.
- Compare the amplitude with the equilibrium extension before assuming the motion is simple harmonic throughout.
- For energy questions, include kinetic, gravitational and elastic terms and use one zero level for height.
CHECK YOURSELF
A mass of 2 kg hangs on a spring of natural length 0.5 m and modulus 100 N. Find the period of small oscillations.
Show a hint
ω² = λ/(ml).
Show the answer
ω
²
=
1
0
0
/
(
2
×
0
.
5
)
=
1
0
0
,
s
o
ω
=
1
0
r
a
d
/
s
a
n
d
T
=
2
π
/
1
0
=
0
.
6
2
8
s
.
Measured from equilibrium the weight cancels, leaving acceleration = −(λ/ml)x: simple harmonic with ω² = λ/ml.
A spring stays simple harmonic throughout; a string goes slack above its natural length, so compare the amplitude with the equilibrium extension.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Prove that a mass on a spring moves with simple harmonic motion and find ω.
- Locate the centre of the oscillation at the equilibrium position.
- Recognise when a string goes slack and the motion stops being simple harmonic.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.