This problem involves the simple harmonic motion of vertical springs.

We need to know the relationship between position, velocity, and acceleration to make the required manipulations to the given equation(s).

**(a)**

We know that y'' = (d^{2}/dt^{2})y = (d/dt)[(d/dt)y]

(d/dt)y = v = y'

An object is attached to a spring hanging from the ceiling. The object undergoes simple harmonic motion modeled by the differential equation my′′ + ky = 0, where y(t) is the height of the object (relative to its equilibrium position) at time t, m is the mass of the object, and k is the spring constant.

(a) Write the phase plane equivalent of the differential equation. The equation should be in terms of y, v, and dv/dy, where v = y'.

_____________

(b) Integrate the phase plane equivalent equation with respect to y to find an equation relating y to v. Write it in the form

KE + PE = E

where KE denotes kinetic energy, PE denotes potential energy, and E denotes total energy.

___________ = E

(c) Suppose the mass is 4 kg and the spring constant is 2 kg/s^{2}. If the spring is initially stretched 4 meters, held, and released, then determine the total energy and write the resulting equation describing the trajectory of the object in the phase plane.

_____________

(d) Graph the trajectory of the object in the phase plane.

(e) The solution can be expressed in the form y = c_{1} cos(ωt) + c_{2} sin(ωt). Use the assumptions of part (c) to determine the values of c_{1} and c_{2}, and ω then use the equation of part (c) to determine ω. Write the expression for y, describing the position of the object at any time.

y = __________

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Our tutors have indicated that to solve this problem you will need to apply the Simple Harmonic Motion of Vertical Springs concept. You can view video lessons to learn Simple Harmonic Motion of Vertical Springs. Or if you need more Simple Harmonic Motion of Vertical Springs practice, you can also practice Simple Harmonic Motion of Vertical Springs practice problems.