Forced oscillation, Equation of Motion and verifying solutions

[sm1t]

Hi, Been registered for a while here, but this is my first post, been using the forum as more of a resource before. I am going through some past papers, but I am faltering at 1 question.

Homework Statement

A particle moving along the x-axis with velocity v experiences a resistive force –bv, but no spring-like restoring force, where b is the friction coefficient.

(a) Write down the equation of motion. [3]
$$\ddot{x}= -(b/m)v$$

(b) Show that the equation of motion is satisfied by
$$x(t) = C - (v_{0}/\gamma)e^{-\gamma}$$

where , m is the mass of the particle, and C and $$\gamma=(b/m)$$ are free parameters. [3]

Not the quickest with latex so the answer was to just differentiate twice, you can then see they are equivalent.

(c) At the particle is at rest at t=0 x=0. At this instant a driving force is switched on $$F = F_{0}cos(\omega*t)$$ what is the equation of motion for t > 0 ? [2]


$$\ddot{x}= (F_{0}/m)cos(\omega*t) -(b/m)v$$

(d) Show that, when both forces are present, x(t)= A*cos(ωt−δ) is a solution to the equation of motion with appropriate choice of A and δ. Find A and δ .
We are also told that [cos(δ) = 1/(rootof 1 +tan^2(δ)] and sin(δ) = tan(δ)/(rootof 1 +tan^2(δ)]

Again I try to differentiate through but I come to a block, I use the 2 above identities but doesn't help me? I know I must be missing something simple or just not seeing it.


Any help much appreciated.

 

[Quantumpencil]

If you haven't already, try verifying it by using complex exponentials rather than trig functions rather than trig functions. It's usually much easier.