The motion of a particle connected to a spring is described by x = 10 sin (π*t). At what time (in s) is the potential energy equal to the kinetic energy? a. 0b. 0.25c. 0.50d. 0.79e. 1.0

t = 0.25 seconds (option b)

Explanation:

Assuming that there is no energy loss due to friction, then the total energy E of the system is conserved and is equal to the sum of kinetic K and potential energy V:

E = K + V

where

V = 1/2*k*x², k = spring constant

K = 1/2*m*v², v = velocity

but x = 10 sin (π*t)

when the particle reaches its maximum amplitude (x=L=10), it has no velocity (can be proven by finding v=dx/dt at t=1/2 for example), then V=0 and E=K max, then

E=Kmax = 1/2*k*L²

when the kinetic energy is equal to the potential energy K=V, then

E=K+V = K+K=2K = 2*1/2*k*x² = k*x²

since also E=1/2*k*L²

1/2*k*L²=k*x²

x = L/√2 = 10 / √2

then

x=10 sin (π*t) = 10 / √2 → t = (1 / π) [sin ⁻¹ (1/√2) ] = 0.25

therefore

t = 0.25 seconds