Calculation Example – Simple harmonic vibration part 2

A single point with mass m is vibrating harmonically. The displacement x from point of balance is governed by the differential equation

where ω is a positive constant.

b) What is the speed, the acceleration, and the force at the mass as a function of time t and constants A, B.

Solution

The derivative of x(t)=A sin⁡ωt+B cos⁡ ωt

is the speed       u(t)=x'(t)=Aω cos⁡ωt-B ωsin⁡ωt 

And the derivative of speed is the acceleration

a(t)=u'(t)=x''(t)=-〖ω^2 Α sin〗⁡ωt-ω^2 B cos⁡ωt

The force at the mass is