A calculation of the period of oscillation for systems that exhibit harmonic motion has a range of difficulty as varied as the number of systems that are described by it. Performing the calculation from first principles for even simple systems often requires preliminary knowledge of basic concepts in differential equations and undergraduate-level physics. The following steps will outline the calculation for the period of oscillation of a mass attached to a spring with one degree of freedom, which, when disturbed, oscillates between compressions and extensions of the spring over a frictionless surface.
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Draw a free-body diagram of the problem. Since the spring (and therefore the mass attached to it) only moves in one dimension, we know that just one variable (x) will be present in the equation of motion. The rest of the quantities that show up are constants.
Write the equation of motion. It will follow from Newton's second law of motion, which tells us that the change in momentum p with respect to time t is equal to the force F responsible for the change (F = dp/dt). For a mass m that is undergoing spring-driven oscillations, Hooke's Law tells us that F = -kx, where k is the spring constant and x is the displacement. Equating the two laws, we get dp/dt = -kx. Putting in all terms of x, remember that momentum is just the mass times the first derivative of x with respect to time (p = m*dx/dt), so the change in momentum is actually the mass times the second derivative of x: dp/dt = m(d^2x/dt^2). Altogether, we have: m(d^2x/dt^2) = -kx.
Solve the equation of motion for x. This can be done intuitively by remembering that the only types of solutions that return equivalents of themselves (save for a factor of -1) after differentiating twice are sinusoidal. In this case we have x = Asin(wt), where A is the amplitude and w is the angular frequency.
Substitute the solution found in Step 3 back into the equation of motion to solve for the angular frequency. For our current example the math appears like this: m(d^2x/dt^2) = -kx = -mw^2(Asin(wt) = -k(Asin(wt)). Cancelling out like terms, we get mw^2 = k --> w = sqrt(k/m), where "sqrt" means "the square root of."
Write the equation for the period T dependence on the angular frequency w: T = 2pi/w. Substitute what w is equal to to get T in terms of what is known: T = 2pi/sqrt(k/m).
Tips and warnings
- Problems of this nature that do not require the use of differential equations should only require Steps 4 and 5 to solve. The general procedure would be to find the angular frequency w from whatever information is known and then to use it to calculate the period T.
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