Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest.
Suppose you push the mass to a position y=0 units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is
y = y_0 cos (t squareroot k/m),
where k > 0 is a constant measuring the stiffness of the spring (the larger the value of k, the stiffer the spring) and y is positive in the upward direction.
Use equation (2) to answer the following questions.
- Find dy/dt, the velocity of the mass. Assume k and m are constant.
- How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring?
- How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (k is increased by a factor of 4)?
- Assume that y has units of meters, t has units of seconds, m has units of kg. and k has units of kg/s^2. Show that the units of the velocity in part (a) are consistent.