Physics · Ch 10 — Oscillations
SIMPLE HARMONIC MOTION (SHM)
SIMPLE HARMONIC MOTION (SHM)
Simple Harmonic Motion (SHM) is a special, particularly important type of oscillatory motion in which the acceleration (or the net force) on the particle is always directly proportional to its displacement from a fixed point, and is always directed towards that fixed point. In one dimension, if is the displacement of the particle from the mean position and its acceleration, then , written with a negative sign as , where is a constant (dimension ) measuring the acceleration produced per unit displacement. Multiplying through by the particle's mass and invoking Newton's second law gives the force law of SHM, , where is the force constant (force per unit displacement, SI unit N m). The minus sign is essential: it says the force (or acceleration) always points opposite to the displacement, i.e. back towards the equilibrium position, so it is called a restoring force. Because the force always pulls the particle back towards the same central point, it is described as a central, attractive force, with the equilibrium position as its centre of attraction. In three dimensions the same law is written with vectors as . …
What this figure shows. A block attached to a horizontal spring is shown at its equilibrium position, and then displaced to either side of it. The equilibrium position is marked as the single point about which the block oscillates symmetrically; when the block is pulled to the right the spring pulls it back to the left, and when pushed to the left the spring pushes it back to the right, illustrating visually why the restoring force always points from the current displaced position back towards th …
What this figure shows. A straight line is plotted through the second and fourth quadrants of a force-versus-displacement graph, passing through the origin with a negative slope. Because the restoring force F_x = -kx is a linear function of displacement x with a negative constant of proportionality, positive displacement always corresponds to negative force (and vice versa), which is exactly why the line lies only in the second and fourth quadrants; measuring the slope of this line lets one read off the numerical value of the force constan …
Worked out. The question asks which of four functions represent SHM: (i) x = A sin(wt) + B cos(wt), (ii) x = A sin(wt) + B cos(2wt), (iii) x = A e^(iwt), and (iv) x = A ln(wt). The test applied throughout is whether the function's second time-derivative works out to exactly -w^2 times the function itself, which is the defining differential equation of SHM. Differentiating (i) twice gives d^2x/dt^2 = -w^2 x exactly, so it represents SHM. Differentiating (ii) twice gives a mix of -w^2(A sin wt) and -4w^2(B cos 2wt) terms that do NOT combine into -w^2x, so it does not represent SHM (the two different angular frequencies wt and 2wt break the proportionality). For (iii), since i^2 = -1, differentiating the complex exponential twice returns -w^2 times the original function exactly, so it does represent SHM. For (iv), differentiating ln(wt) twice gives a te …