A particle of mass `0.2 kg` undergoes SHM according to the equation `x (t) = 3 sin (pi t + pi//4)`.
i. What is the total energy of the particle if potential energy of zero at mean position ?
ii. What are the kinetic and potential energies of particle at time `t = 1 s`?
iii. At what time instants is the particles energies purely kinetic?

Explanation

Camparating the given equation with `x (t) = A sin (omega t+ phi_(0))`. We get `A = 3m, omega = pi rad//s. phi_(0) = pi//4`
`T = (2 pi)/(omega) = 2 s`
`x (0) = 3 sin pi//4 = 1.5 sqrt2 m`
`dx//dt = v(t) = 3 pi cos (pi t + pi//4)`
`implies v(0) = 3 pi//sqrt2 m//s`
i. Total energy `= (1)/(2)KA^(2) = (1)/(2) m omega^(2) A^(2) = 1//2 (0.2) (pi)^(2) (3)^(2) = 0.9 pi^(2)`
ii. At `t = 1, x(t) = 3 sin (pi + pi//4) = - (3)/(sqrt2)m`
`v (t) = 3 pi cos (pi + pi//4) - (3 pi)/(sqrt2)m//s`
`K = (1)/(2) mv^(2) = (1)/(2) (0.2) ((9pi^(2))/(2)) = (9pi^(2))/(20) J`
`u = (1)/(2) kx^(2) = (1)/(2) m omega^(2) x^(2)`
`= (1)/(2) (0.2) (pi)^(2) ((9)/(2)) = (9)/(20)pi^(2) J`
iii. Energy is purely kinetic at mean position, i.e., when `x = 0`
using `x (t) = 3 sin (pi t + pi//4)` we have
`0 = 3 sin (pi t + pi//4)`
`pi t + pi//4 = 0, pi, 2 pi, 3 pi`,...
`implies t = (3)/(4) s, (7)/(4) s, (11)/(4) s`...
At these time instant , partical crossec origin and hence its energy is purely kinetic.