Question

A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and s is in inches s(t)=−2cos(πt+ π/6)

For t>0, determine the first two times at which the velocity of the spring is zero. Type answer in form t=c. Separate multiple answers with a comma, such as t=1,t=−1

Answer 1

In simple harmonic motion, the velocity of the spring is zero at the extremes of its motion, which are the positions where the displacement is maximum. To find the times at which the velocity is zero, we can set the velocity function equal to zero and solve for t. The first two times at which the velocity is zero are t = -1/6 and t = 5/6.

Step-by-step explanation:

In simple harmonic motion, the velocity of the spring is zero at the extremes of its motion, which are the positions where the displacement is maximum. In this case, the displacement function is given by s(t) = -2cos(πt + π/6). To find the first two times at which the velocity is zero, we need to find the values of t that satisfy v(t) = 0.

The velocity function can be found by differentiating the displacement function with respect to time: v(t) = -2πsin(πt + π/6). Setting v(t) equal to zero, we get -2πsin(πt + π/6) = 0. Simplifying, we have sin(πt + π/6) = 0.

Solving sin(πt + π/6) = 0, we can use the fact that the sine function is zero at integer multiples of π. Therefore, we have πt + π/6 = 0, πt + π/6 = π, and πt + π/6 = 2π. Solving these equations for t, we get t = -1/6, t = 5/6, and t = 11/6. Therefore, the first two times at which the velocity of the spring is zero are t = -1/6 and t = 5/6.