Using differential equation of linear S. H.M, obtain the expression for (a) velocity in S. H.M., (b) acceleration in S. H.M.

Let , ,  and denote three constants (with the requirement that .) The following assumes that the mass of this object is . Assume that denotes the position of the object at time .

(a) .

(b)

Explanation:

The differential equation for a simple harmonic motion might take the following form:

.

The minus sign on the right-hand side highlights the fact that the displacement and acceleration of the object should be in opposite directions.

Notice how this equation is in the form of a homogeneous second-order ODE:

Let be a constant. One possible solution to this homogeneous second-order ODE would be in the form , such that whereas .

Substitute into the original ODE to obtain:

.

Rearrange the equation and solve for .

Notice that . Hence, it must be true that . Solve for given that :

, where is the imaginary unit.

The two particular solutions for the ODE would be:

and .

Apply Euler's Formula to rewrite both solutions in terms of trigonometric functions:

.

.

The general solution would be in the form:

,

Where and are constants (not necessarily real numbers.)

Since position is supposed to assume a real value for any real , set to a multiple of such that the general solution is real-valued:

.

Differentiate to obtain general expressions for velocity (first derivative) and acceleration (second derivative.)

probably not if net force is zero

talk back to him about the sport and show him your expertise or leave the room