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# Simple Harmonic Motion

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## Simple Harmonic Motion

The time taken for an oscillating object to complete one full oscillation is called the **time period, T.** It is measured in seconds.

If a number of oscillations are involved we can work out the time period by dividing the total time taken by the number of oscillations completed:

The **frequency, f,** of oscillations is the number of oscillations undergone in one second, and is measured in **hertz (Hz).**

**The frequency and the period can therefore be related as:**

The **displacement** of an oscillating particle is the distance the particle has been moved from its equilibrium position.

The **amplitude** of an oscillation is the maximum displacement of the vibrating object from the equilibrium position (its usual position).

* Note:* Always check the x-axis on the graph, as it is easy to confuse wavelength and time period on diagrams!

**Simple Harmonic Motion (SHM)** is a particular type of oscillation. It is useful because its time period stays the same even when its amplitude changes. We'll come to the full definition later!

**Lets think about a simple example of shm to work out the relationship between displacement, velocity and acceleration:**

Now remember that displacement, velocity and acceleration are all vectors, and as a result, direction is important. Let's choose anything in the **up-wards** direction to be **positive,** anything **downwards** to be **negative.** (If you decide to do the opposite, it doesn't matter - just stick to your choice.)

**If we set this system oscillating by lifting the mass and letting it go, then the system starts with:**

**Maximum positive displacement** (because it's above the middle).

**Zero velocity** (it's not moving at the first instant).

**Maximum negative acceleration** (because it is about to start moving down).

The interaction below shows how velocity and acceleration change in simple harmonic motion. It shows the relationship between velocity and acceleration. **Click "next" to see each part of the motion...**

The displacement, velocity and acceleration of the mass are related as shown above. To draw these, think about what the object is doing at each point as it oscillates from the start position described above.

As it passes through the equilibrium position on the way down it's at maximum speed down (negative), its displacement is zero and because the spring is at its equilibrium position, there is no resultant force on the mass so it is not accelerating.

At the bottom, the mass stops momentarily as it changes direction, so velocity is zero. The displacement is a maximum in the negative direction, so the acceleration is a maximum in a positive direction as the spring tries to shorten again.

The important point to note is **the phase difference** between these three variables...

**1.** The **velocity, v,** is zero where there are stationary points at the peaks and troughs of the displacement graph and the velocity is a maximum when the displacement is zero. **(Don't forget the gradient of the displacement graph will equal velocity.)**

**2.** The displacement and acceleration graphs are 180 degrees out of phase and therefore look like a mirror image of each other in the time axis. **(Don't forget the gradient of the velocity graph will equal acceleration.)**

**Definition of Simple Harmonic Motion:**

All of the above leads us to the formal definition of shm:

**A body is undergoing SHM when the acceleration on the body is proportional to its displacement, but acts in the opposite direction.**

Acceleration is proportional to displacement

a α - s

It's also important to note that for SHM, the time period of the oscillations is constant and doesn't change even if the amplitude is changing.

**There are two common examples of simple harmonic motion:**

Where m = mass (kg)and k = spring constant (Nm-1) | Where L = length of pendulum (m)g = acceleration due to gravity (ms-2) |

SHM is used to explain the behaviour of atoms in a lattice, which oscillate like masses on springs.

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