Determining the equation for simple harmonic motion of an object on a spring Itai Buxbaum

LD22/1/15

Learning Object

The situation: In your sleep, you dream that you have invented a

special, frictionless spring mechanism, where no energy is being lost to friction (thermal energy).

As you wake up, you remember that the 2.0 kg object on the spring would move 12 meters from the equilibrium point, then return, and continue 12 meters past the equilibrium point in the other direction! It took 6 seconds for this period to complete its cycle.

Luckily remember that in PHYS 101 you learned about SHM and now you decide that you want to graph the functions for velocity displacement and acceleration!

Game plan: what do we need to figure out? We will need to find:

The period, T and frequency, f the angular frequency ω A graph of the displacement of the object A graph of the velocity of the object And a graph of the acceleration of the

object

What do you remember? Lets parse what we know into physics terms…

The mass of the object was 2 kg

m=2 kg

The max displacement was 12m

Amplitude, A= 12

It took 6 seconds to complete the period

Period, T= 6

No energy was lost due to friction

Cool!

Step 2. lets get to it!

Lets start with angular frequency ω, the magnitude of the angular velocity!

We know that T, period is 6 seconds so

Step 3. Put it into a function: The displacement function for Simple

Harmonic Motion looks like this: X(t)=Acos(ωt+ϕ) ϕ will be zero because we will start x(0)

with maximum displacement. Lets plug in what we have figured out

and then graph it:

Graph displacement X(t)=Acos(ωt+ϕ)

To start the graph, note the amplitude and the period first, then draw the cosine curve , starting at maximum displacement when t=0 and finishing its cycle every 6 seconds

Now for velocity: We can differentiate the displacement

function to get the velocity function.

You graph this the same way. The T stays the same, and the amplitude is or about -12.56

For acceleration: We can differentiate the velocity

function to get the acceleration function.

Conclusion:Displacement

Velocity

Acceleration:

When the displacement is at its maximum, the velocity is zero, and the acceleration is at its maximum in the -, and is going towards zero. this makes sense.