Definition of SHM Objectives Define simple harmonic motion. Select and apply the defining equation for simple harmonic motion. Use solutions to the equation.

Oscillating objects should continue to oscillate with a uniform period. This can be verified in several different experiments. A swinging pendulum A bouncing spring An oscillating trolley All 3 experiments can be shown to oscillate uniformly with the Data Logger running the LogIT lab software set to measure the SHM period. In all cases it can be shown that the restoring force is in the opposite direction to the position relative to the rest/equilibrium position.

Experiment 2 Experiment 3 – could be done on the air track! Experiment 1 Activity 6

Change the mass and measure the period of Oscillation as the mass is changed. Remember to calculate the period accurately Log It software Using the light gates and the Log It software. You will need to set the software up to measure the SHM Period. Select the “Timing” option from the menu and then “SHM Period”. You can then tell it to time at any sensor and just have one attached. You should check the timing by using stopclocks and a fudicial marker. Use a pencil or similar attached to the bottom of the masses to cut the gate cleanly at the midpoint.

Just the same as we should be able to understand the equation for the period of a pendulum, we also need to be able to do it for an oscillating spring. SO……. l Consider a spring of length “l” and spring constant “k” Bouncing masses - theory

l + e When a mass “m” is hung from the spring it has an extension “e” due to the weight pulling downwards - “mg”. W = mg Mass “m” T = ke This causes a restoring force in the spring, or Tension. These two forces are in equilibrium unless the mass is moved! Therefore mg = ke

l + e + x W = mg T = k(e+x) l + e W = mg T = ke x upwards When the spring is pulled down a distance “x” before release the forces no longer balance. There is a resultant force upwards. Net force upwards = k(e+x) - mg

Using Newton’s second law we can see that the upward force must be equal to “ma”. Therefore: - {k(e+x) - mg} = ma And since mg = ke We get -kx = ma In other words the restoring force is in the opposite direction to the displacement from the rest position! sinusoidal sine or cosine This type of movement is called sinusoidal as it varies like a sine or cosine wave form.

Since we knew a = -cx It was later proved that in this case the constant c =  2 a = -  2 x = -(2  f) 2 x Therefore a = -  2 x = -(2  f) 2 x two possible solutions So the motion known as SHM can be described using two possible solutions to the equation: x = x o cos(  t) = x o cos( 2  f) t x = x o cos(  t) = x o cos( 2  f) t OR x = x o sin(  t) = x o sin( 2  f) t

We can go on to prove a link between the period and other factors for a spring from these equations. This will ultimately show that the period of a spring only depends on the mass and spring constant NOT the amplitude of the motion! The same can be shown for a pendulum or the oscillating trolleys! Using SHM equation a = -  2 x = -(2  f) 2 x -kx = -mx(2  f) 2 k/m = (2  f) 2

Simple Harmonic Motion If a particle oscillates about a point with SHM, its acceleration is proportional to its displacement from that point and directed towards that point. Acceleration / ms -2 Displacement m Constant of proportionality minus The minus sign indicates that the acceleration is always in the opposite direction to the displacement i.e. towards the equilibrium position.

The negative slope is due to the negative sign and indicates the direction of the acceleration. That is, the acceleration and displacement are always in opposite directions. This equation is the equation of motion for SHM. Therefore :-