Filip B. answered • 07/22/15
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Hi there! To answer this question you need to recall the key features of a sine wave, it is given to us in the form:
y(t) = Asin(ωt)
Where A is the amplitude in meters, ω is the angular frequency, and t is the time in seconds.
Using this and a few other things, we can figure out all of the things needed for this question. Firstly, the amplitude is given to us straight from the equation. We can find out the frequency because it is inversely related to the period. Remember that the angular frequency (ω) is equal to 2π/T where T is the period. Using this you can find both the frequency and the period.
The spring constant is also related to the angular frequency, in that ω = √(k/m) where k is the spring constant, and m is the mass.
In order to find the max velocity, we need to get the velocity function from the position function. To do this we need to take the derivative with respect to time, so we end up with a velocity equation that looks like:
v(t) = Aωcos(ωt)
And we know that this function will be a maximum when the cosine portion is maximum, so we can just set cos(ωt) equal to 1, and then the velocity function will be maximum.
The total energy in a system like this is a combination of its kinetic and potential energy. We know that as the mass passes the center of its oscillation, it has maximum kinetic energy and no potential energy, and that when the mass is at the peak or trough of its oscillation, it has maximum potential energy and no kinetic energy. So if we calculate the energy at one of these points, we only need to consider one form of energy. The easiest thing to do, since we already know the maximum velocity, is to find the kinetic energy when velocity is maximum, as this will be equal to the total energy of the system.
So if we do K.E. = 1/2 m Vmax2 then we end up with 1/2 m (Aω)2
I hope this helps your understanding of the various formulas needed for this kind of situation!
