Lab 9: Simple Harmonic Motion, Mass-Spring Only 3 more to go!! The force due to a spring is, F = -kx, where k is the spring constant and x is the displacement from equilibrium. F = - kx +x This type of force is called a “restoring force” because it always acts to restore the system to equilibrium. If we pull on the spring-mass system and let it go it will oscillate back and forth. If we make a plot of the spring’s position vs time we get: As you can see this is a cosine function.

It turns out we can write the displacement as a function of time, t: x = x MAX cos(  t) The velocity can be written as: V = -  x MAX sin(  t), The acceleration: a = -  2 x MAX cos(  t) t – time in seconds x MAX – maximum displacement, or amplitude  - angular frequency, tells us how many oscillations occur per second. For a spring mass system: where k, is the spring constant and m, is the mass Another useful quantity is the period, T. The period tells us how long it takes to complete one oscillation, for a spring mass system:

3 kg Let’s consider a spring-mass system, k= 30 N/m, m = 3 kg. The mass oscillates with an amplitude of 0.1m x MAX = 0.1 m What’s the frequency of osc. ? What is the displacement at t = 0? x = x MAX cos(  t) = 0.1 m cos (0) = 0.1 m What is the max. and min acceleration? a = -  2 x MAX cos(  t), so the max accel. occurs when the cos(  t) term equals 1, when does this happen? This happens when (  t) = 0,  (i.e. at the turning points)

When is the velocity maximum? V = -  x MAX sin(  t) velocity is max. when sin(  t) is max (i.e. equals 1), this happens when (  t) =  /2 What are we doing today? 2 experiments. The first will allow us to measure the spring constant, k, of our spring. You will hang the spring, measure the equilibrium length. Next you will and mass, and measure how much it stretches. Do this several times for different masses. Make a plot of Weight (y-axis) vs Displacement from equilibrium (x-axis)

The next exp, will allow us to find the spring constant, k using, the oscillatory motion of the spring First, put 100g on the end of spring and pull down on the spring slightly. Release and record the time it takes to make 25 oscillation. Calculate the period, T = time/25 Increase the mass and repeat the experiment 5 times Normally, the period T: But this doesn’t account for the fraction,f of the spring that is also oscillating, to account for this we derive a new equation for the period: Where, f is the fraction of the spring that is oscillating, and M S is the mass of the spring If I find the equation for T 2 :

Now let’s plot T 2 (y-axis) vs. mass, m (x-axis):