Question

A mass that weight stretches a spring . The system is acted on by an external force . If the mass is pushed up and then released, determine the position of the mass at any time . Use as the acceleration due to gravity. Pay close attention to the units.

Answer 1

Answer:

x = A cos wt

Explanation:

To determine the position we are going to solve Newton's second law

             F = m a

           

Spring complies with Hooke's law

          F = -k x

And the acceleration of defined by

          a = d²x / dt²

         

We substitute

        - k x = m d²x / dt²

         dx² / dt² + k/m  x = 0

Let's call

           w² = k / m

The solution to this type of differential equation is

           x = A cos (wt + Ф)

Where A is the initial block displacement and the phase angle fi is determined by or some other initial condition.

In this case the body is released so that at the initial speed it is zero

From which we derive this expression

         v = dx / dt = a w sin ( wt + Ф)

         

As the System is released for t = 0 the speed is v = 0

         v = sin Ф = 0

Therefore Ф = 0

And the equation of motion is

         x = A cos wt