Definition

- Simple Harmonic Motion is a type of motion where:
- An object oscillates back and forth about a point of zero displacement at the centre
- The force on the object (and therefore the acceleration of the object) is proportional to the magnitude of the displacement
- The force and acceleration are always towards the point of zero displacement (middle)
- The period of the oscillation is the same regardless of the amplitude
- The oscillation is sinusoidal

Springs Do It

- Masses on springs oscillate with simple harmonic motion because:
- The force provided by the spring is F = -kx
- The magnitude of the force is proportional to the displacement
- The spring constant, k, is the constant of proportionality
- The force, F is in the opposite direction from the displacement / extension of the spring, x, hence the minus sign in the equation
- (The force is what produces the acceleration on the mass)
- If the mass is on a vertical spring, weight acts on it as well as the thrust or tension in the spring
- The point of zero displacement would be where the weight balances the force from the spring, so that if the mass was moved in either direction the forces would no longer be balanced and there would be a resultant force back towards the centre
- If the mass is on a horizontal spring, the point of zero displacement is where the spring is at its normal length
- Moving the mass one way would compress the spring and moving it the other way would extend it, resulting in a force towards the centre
- Because it’s simple harmonic motion, we know that the acceleration is proportional to the displacement (or extension, for a spring) but negative, i.e. a α -x
- The formula booklet says that the constant of proportionality is k/m (this spring constant divided by the mass oscillating), so a = -(k/m)x

Pendulums Do It

- Pendulums oscillate left and right with Simple Harmonic Motion
- The restoring force (force that pulls the mass back to the centre) is provided by the horizontal component of the tension in the string
- Whenever the mass is not directly under the point it’s suspended from, the direction of the tension is diagonal, so there’s a horizontal component
- This component is directed towards the centre, and its size is proportional to the displacement
- The force is approximately F = -(mgs)/L
- F is the force, m is the mass of the object, g is the acceleration due to gravity / gravitational field strength and L is the length of the string
- This equation is only true when the angle is small, because it ignores a bit of trigonometry

Where the Displacement and Time Equation Comes From (in Detail)

- Because the oscillation is sinusoidal, the graph of displacement against time is a sine curve
- The equation of the graph will be based on the form s = sin(θ), where θ is some angle
- s is the symbol for displacement, which can be negative or positive depending on its direction
- The sine of anything is always between -1 and 1, so unless the amplitude of the oscillator being modelled is exactly 1, it’ll need to be included in the equation:
- s = A sin(θ), where A is the amplitude of the oscillator: the maximum distance between it and the centre
- We want an equation for displacement, in terms of time, so θ isn’t much help
- θ can be replaced with 2πft, which is the phase of an object oscillating with frequency f at time t
- Notice the 2π – this equation is in radians. Make sure your calculator is in radians mode!
- The equation becomes: s = A sin(2πft)
- s is the displacement at time t, A is the amplitude, f is the frequency and t is the time

Getting Velocity and Acceleration Equations

- Velocity is the rate of change of displacement (think of distance over time)
- v = ds/dt, so differentiating s with respect to t produces an equation for v
- s = A sin(2πft) is in radians, so it can be differentiated:
- v = 2πfA cos(2πft)
- Furthermore, acceleration is the rate of change of velocity
- The equation for v can be differentiated again to get the equation for a
- a = -4π
^{2}f^{2}A sin(2πft) - It may help you to learn the rules for differentiating sines and cosines (If you study maths, you probably already know them)
- Differentiating a sine makes it a cosine and differentiating a cosine produces a negative sine
- The resulting sine or cosine is multiplied by the 2πf from inside the brackets, as 2πf is the coefficient of the t, which we’re differentiating with respect to
- The equation for a has the 2πf from the v equation multiplied by the 2πf from the differentiation, to make 4π
^{2}f^{2}

Comparing the Equations

- s = A sin(2πft), v = 2πfA cos(2πft) and a = -4π
^{2}f^{2}A sin(2πft) - The frequency of the displacement, velocity and acceleration is the same
- There is a 90° phase difference between displacement and velocity
- There is a further 90° phase difference between velocity and acceleration
- The phase difference between displacement and acceleration is 180°
- The acceleration is proportional to the displacement, but negative – this is because the force and acceleration are always towards the centre, i.e. in the opposite direction from displacement

Data, Formulae and Relationships Booklet

- F = kx is in the booklet. Remember that F is in the opposite direction from x
- You probably won’t need the approximation for the pendulum’s restoring force, which is good, because it’s not in the booklet
- s = A sin(2πft) is in the formula booklet, but with the letter x instead of the letter s
- x = A cos(2πft) is also in the formula booklet. This is used when the timer has been started with the oscillator at maximum positive displacement instead of when it was at zero displacement, causing it to form a cosine curve
- The equations for v and a aren’t exactly in the formula booklet, because they differ depending on whether you’re using sine or cosine for displacement. What is in there, however, is:
- d
^{2}x/dt^{2}= a = -(k/m)x = -(2πf)^{2}x - (That’s telling you that a is the second derivative of displacement with respect to time, proportional to displacement by a constant of proportionality -k/x and equivalent to the displacement multiplied by 2πf twice and made negative)
- The Simple Harmonic Motion equations given in the formula booklet are for spring oscillators and not pendulums

There are two more parts to this post…

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