Question

Consider one dimensional motion of a particle of mass m . It has potential energy U = a + bx2 where a and b are positive constants. At origin (x = 0) it has initial velocity ν0. It performs simple harmonic oscillations. The frequency of the simple harmonic motion depends on

Answer 1

it is given that, U = a + bx²

differentiating with respect to x,

dU/dx = 2bx

we know, F = -dU/dx

so, F = -2bx

from Newton's 2nd law, F = ma = -2bx

or, a = -(2b/m)x

from simple harmonic motion,

a = -w²x

comparing both sides,

w = √(2b/m)

or, 2πf = √(2b/m)

hence, f = 1/2π √(2b/m)

hence, frequency of the simple harmonic motion depends on mass.