At the equilibrium position, the potential energy is zero, and the velocity and kinetic energy are maximized. The kinetic energy at the equilibrium position is equal to the mechanical energy:

    From this equation, we can derive the maximum velocity:

    You won’t need to know this equation, but it might be valuable to note that the velocity increases with a large displacement, a resistant spring, and a small mass.

 

    Summary

 

    It is highly unlikely that the formulas discussed above will appear on SAT II Physics. More likely, you will be asked conceptual questions such as: at what point in a spring’s oscillation is the kinetic or potential energy maximized or minimized, for instance. The figure below summarizes and clarifies some qualitative aspects of simple harmonic oscillation. Your qualitative understanding of the relationship between force, velocity, and kinetic and potential energy in a spring system is far more likely to be tested than your knowledge of the formulas discussed above.

    In this figure, v represents velocity, F represents force, KE represents kinetic energy, and represents potential energy.

 

    Vertical Oscillation of Springs

 

    Now let’s consider a mass attached to a spring that is suspended from the ceiling. Questions of this sort have a nasty habit of coming up on SAT II Physics. The oscillation of the spring when compressed or extended won’t be any different, but we now have to take gravity into account.

 

    Equilibrium Position

 

    Because the mass will exert a gravitational force to stretch the spring downward a bit, the equilibrium position will no longer be at x = 0, but at x = –h, where h is the vertical displacement of the spring due to the gravitational pull exerted on the mass. The equilibrium position is the point where the net force acting on the mass is zero; in other words, the point where the upward restoring force of the spring is equal to the downward gravitational force of the mass.

    Combining the restoring force, F = –kh, and the gravitational force, F = mg, we can solve for h:

    Since m is in the numerator and k in the denominator of the fraction, the mass displaces itself more if it has a large weight and is suspended from a lax spring, as intuition suggests.

 

    A Vertical Spring in Motion

 

    If the spring is then stretched a distance d, where d < h, it will oscillate between and .

    Throughout the motion of the mass, the force of gravity is constant and downward. The restoring force of the spring is always upward, because even at the mass is below the spring’s initial equilibrium position of x = 0. Note that if d were greater than h, would be above x = 0, and the restoring force would act in the downward direction until the mass descended once more below x = 0.