Question

The motion of a particle connected to a spring is described by x=10cos(πt). At what time (in s) is the potential energy equal to the kinetic energy?

Answer 1

That would be when the absolute value of the deflection is sqrt(1/2) of the maximum value. At those times, X^2 is half the maximum value, and so is the potential energy.

That happens when pi*t = pi/4 or 3*pi/4 or 5 pi/4 or 7 pi/4.
t = 1/4, 3/4, 5/4 or 7/4 s.

In this case the period is 2 seconds.

Answer 2

Thank you so much!