A 50g mass is connected to a spring of spring constant k and is oscillating with

Question

A 50g mass is connected to a spring of spring constant k and is oscillating with the spring on a frictionless table. The other end of the spring is fixed to a vertical wall as shown in the figure. Consider a certain time to be t=0s when you observe that the mass is at x=2.73 cm (from the equilibrium position of x=0; note that this is NOT Xmax) and make a measurement that it is moving to the right at a speed of 47.02 cm/s at this instant. You also observe that the mass is at this position, with this velocity in the same direction, next at t=0.2s.

a) Find an expression for the position as a function of time, explain the physical significance for each of the terms there in and find numerical values for each of the terms there in and find numerical values for each of the terms/variables/symbols in your expression for x(t).

b) How much energy is stored in the spring-mass system when the mass is 1.7 cm from its equilibrium position?

Explanation / Answer

We can use x = A sin (w t + phi)   for simple harmonic motion (SHM)
You can see that T (period) = .2 sec
Then f = 1 / T = 5 and w = 2 * pi * f = 31.4 for the angular frequency
2.73 = A sin phi      at t = 0
Since v = A w cos (w t + phi)
47.02 = A * 31.4 * cos phi        at t = 0
2.73 / 47.02 = tan phi       dividing equations
phi = 3.323 deg = .3.323 * 2 * pi / 360 = .058 rad
Then A = 2.73 / sin 3.323 = 47.1 cm
And our equation for SHM becomes
x = 47.1 sin (31.4 t + .058)     Since w * t is in radians
Also w = (k / m)^1/2    and k = m * w^2
E = 1/2 k x^2 = 1/2 * m * w^2 * .017^2    to get answer in Joules
E = 1/2 * .05 * 31.4^2 * .017^2 = .00712 J

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