If the phase angle for a block-spring system in SHM is ϕ and the block's position is given by x = xm cos (ωt + ϕ), what is the ratio of the kinetic energy to the potential energy at time t = 0? (Use any variable or symbol stated above as necessary.)

K. E/P. E = m/k tan²φ x ω²

Explanation:

The given position of block x = x₀ cos (ωt + φ)

The velocity of block v = dx/dt = - x₀ sin (ωt + φ) x ω

The kinetic energy = 1/2 mv² = 1/2 m x₀² sin² (ωt + φ) x ω²

The potential energy of spring = 1/2 k x², where k is the spring constant

Thus P. E = 1/2 x k x x₀² cos² (ωt + φ)

When t = 0

K. E = 1/2 m x₀²sin²φ x ω²

P. E = 1/2 k x₀² cos²φ

Dividing these, we have

K. E/P. E = m/k tan²φ x ω²