Simple Harmonic Motion

• Analyse simple harmonic motion mathematically
• Make use of Hooke's Law to study the behaviour of springs

Mandatory Experiment:

Investigation of the relationship between period and length for a simple pendulum, and hence calculation of g.

Hooke's Law

Hooke's Law states that when a force is needed to extend or compress a spring by some distance, the restoring force is proportional to the displacement.

$F = - ks$ (F&T p 57)

Where k is called the elastic constant or the spring constant. The "minus" signifies that the restoring force and the extension (or compression) are opposite in direction. (can be ignored for calculations). The most common example of Hooke's Law is a stretched string.

An object is said to be moving with Simple Harmonic Motion if;

• its acceleration is directly proportional to its displacement from a fixed point in its path
• its acceleration is directed towards that point.

$a ∝ s$ $a = -ω²s$

$ω$ — constant

Examples of objects moving with SHM:

• A mass attached to the end of a spring vibrating up and down
• A pendulum swinging through a small angle
• A prong on a vibrating tuning fork

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A body is moving with SHM:

Explain what is meant by each of the following terms:

• Equilibrium position: point where the acceleration and the overall force on the body is zero
• Amplitude: the maximum displacement from the equilibrium position (acceleration greatest here)
• Cycle or oscillation: the motion of the body from a point until it is back in the same point moving in the same direction
• The periodic time or the period: time taken for one cycle
• The frequency: the number of cycles per second

To show that any object that obeys Hooke's Law will also execute SHM. The aim here is to show that Hooke's Law is a subset of S.H.M.

Q. A mass at the end of a spring is an example of a system that obeys Hooke's Law. Give two other examples of systems that obey Hooke's Law. Stretched elastic, stretched spring, pendulum, oscillating magnet, springs on a car, object bobbing on water waves.

Periodic Time (T) for an object undergoing SHM

As with waves, the periodic time is the time taken for one complete oscillation.

$a = -ω^2 s$

Is the equation of motion of a body moving with simple harmonic motion.

Write down the formula for the period of the motion.

$T = \frac{2π}{ω}$

Also

$T = 1/f$

f = Frequency

Notes on calculations:

When using the F = -k s expression for Hooke's Law, s represents the extension, i.e. the distance between the new length and the original (natural) length.

However when using the expression for simple harmonic motion a = (-ω² s) s represents the distance between the new length and the equilibrium position.

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The Simple Pendulum

A simple pendulum consists of a light string with a mass at the end.

Periodic Time of a Simple Pendulum

Relationship between the period and the length of a simple pendulum: the square of the period is directly proportional to its length.

2014 Question 12 (a)

Sketch a velocity-time graph of the motion of the object undergoing simple harmonic motion

Solution

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Investigation of the relationship between periodic time and length for a simple pendulum and hence calculation of g

Apparatus

Pendulum bob, split cork, ruler, string and timer

Diagram

Procedure

• Place the thread of the pendulum between two halves of a cork or between two coins and clamp to a stand
• Measure length (l) from fixed point to top of bob (using metre stick)
• Measure the diameter of the bob using a digital calipers and calculate the radius (r)
• Total length = l + r
• Set the pendulum swinging through a small angle (<10°)
• Measure the time t for thirty complete oscillations. Divide this time t by thirty to get the periodic time T
• Repeat for different lengths of the pendulum
• Draw a graph of T² against length l and use the slope to calculate a value for g (g = 4π²/slope).

Sources of error/precautions

• Measuring from the bottom of the split-cork to the centre of the bob
• Do not cause the pendulum to swing through an angle greater than 10 degrees
• Ensure that the pendulum swings in one plane only - avoid circular movements
• Use a long pendulum as much as possible to minimise percentage errors associated with measuring the length and also the time