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 y₀ 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

here 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.
a. Find dy/dt, the velocity of the mass. Assume k and m are constant.

b. How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring?

c. 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)?

d. Assume that y has units of meters, t has units of seconds, m has units of kg, and k has units of kg/s². Show that  he units of the velocity in part (a) are consistent.

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