Question

1. A certain spring is compressed 0.2 metres from its natural length by a force of 0.02 newtons. A 0.1 kilogram mass is attached to this spring. There is no damping, and the mass is acted on by an external force of 0.05 cos(0.8 t) newtons, where t is measured in seconds. At t = 0, the mass is released, at rest, from its rest (equilibrium) position.

   1. (a) Set up and solve the initial value problem for the displacement x(t) of the mass from its rest (equilibrium) position, where x is measured in metres. Express x(t) as a sum of sinusoidal (i.e. sine or cosine) functions of t.

   2. (b) Express x(t) from part (a) as a product of sinusoidal functions of t, and identify the two “envelope” functions that x(t) oscillates between.

Answer 1

First, it is neccesary to obtain the value of the spring constant:

Then, according to Newton's second law (without damping force):

a.

The differential equation will be:

which is equivalent to:

where initial conditions are:

Notice that the differential equation is a non-homogeneous equation. Then, general solution is the sum of a homogeneus and particular solution.

Solving the homogeneus differential equation:

A particular solution for this equations is a cosenoidal function, so we can define that the particular solution as:

Then:

Finally, general solution:

Using initial conditions:

Later:

b.

Applying trigonometric identities for products of cosine functions:

The "envelope" functions are the sine function with the lowest frecuency. Then:

The graph of solution and envelope functions is the following: