Question

1) Sinusoidal Motion Properties in Spring Mass System- A 200g mass hangs from vibrating spring at lowest point of 3cm above table and at it's highest point at 12cm above table. It's oscillation period is 4seconds. Determine the following:

a. The spring constant in terms of T (period)

b. The maximum velocity magnitude and maximum acceleration magnitude

c. The velocity magnitude at 10cm above table

d. The vertical position, velocity magnitude and acceleration magnitude at 5 seconds

Answer 1

PART A:

The period of oscillations is

The angular frequency is given by

The relation between the angular frequency and the spring constant is

So, the spring constant as a function of T is given by

And for T=4 s.

PART B:

The distance between the lowest and the highest point is

So, the amplitude of the oscillations is given by

The equation for simple harmonic oscillations is given by

The maximum velocity magnitude is

And the maximum acceleration magnitude is

PART C:

The position we are interested in is 10 cm above the table.

The distance between this point and the equilibrium point is

Putting it in

So, the velocity magnitude at this time is

PART D:

At t = 5 s.

The position is given by

The velocity is given by

And the acceleration is given by

And the magnitude is 0.111 m/s².