Question

A mass attached to a spring is pulled toward the floor so that its height above the floor is 10 mm (millimeters). The mass is then released and starts moving up and down reaching maximum and minimum heights of 20 and 10 mm , respectively, with a cycle of 0.8 seconds.
a) Assume that the height h(t) of the mass is a sinusoidal function, where t is the time in seconds, sketch a graph of h from t = 0 to t = 0.8 seconds. t = 0 is the time at which the mass is released.
b) Find a sinusoidal function for the height h(t).
c) For how many seconds is the height of the mass above 17 mm over one cycle?

Answer 1

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a) The mass is released at t = 0 when h is minimum. Half a cycle later h reaches its maximum and another half a cycle it reaches its minimum again. Hence over one cycle, h varies with t as follows: b) According to the graph obtained in part a), h(t) could be modeled by a cosine function shifted (translated) vertically up and horizontally to the right. Hence 

h(t) = a cos[ b(t - d) ] + c 

Let hmax be the maximum value of h and hmin be the minimum value of h. Hence 

|a| = (hmax - hmin) / 2 = (20 - 10) / 2 = 5 , a = ~+mn~5 

c = (hmax + hmin) / 2 = (20 + 10) / 2 = 15 a natural or artificial chemical compound consisting of large moleculesa natural or artificial chemical compound consisting of large molecules

Period = 2π / |b| = 0.8, hence b = ~+mn~ 2.5π 

We use a = 5 and b = 2.5π. The shift of the cosine function is to the right and equal to half a period. Hence d = 0.4 

h(t) = 5 cos[ 2.5π(t - 0.4) ] + 15 

Check that h has a minimum at t = 0: h(0) = 5 cos[ 2.5π(0 - 0.4) ] + 15 = 5 cos( -π ) + 15 = 10 

Check that h has a maximum at t = 0.4: h(0) = 5 cos[ 2.5π(0.4 - 0.4) ] + 15 = 5 cos( 0 ) + 15 = 20 

c) Below is shown the graph of y = h(t) and y = 17. We first need to find t1 and t2 which are the values of t for which h(t) = 17 by solving the equation 

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