# Question based on oscillatory motion

I came across this question in harmonics= A small glass bead of mass m initially at rest starts from a point at height h above the horizontal and rolls down the inclined plane AB as shown. Then it rises along the inclined plane BC. Assuming no loss of energy, the time period of oscillation of the glass bead is:

I first ascertained that we can divide the whole course of motion into two parts of oscillatory motion. But when I was calculating the force we get it as constant and does not vary with displacement. What am I missing?

You're right to identify that the forces involved here are not "restoring forces" like you would see for a conventional harmonic oscillator situation. The force is always downwards with constant strength $$mg$$.

What does happen is the ball reaches the bottom of the first slope with some velocity $$v$$ (which you can work out with standard $$suvat$$ equations) and then starts rising up the second slope with this initial velocity $$v$$. Because now $$v$$ is opposite the direction of force (still $$mg$$), the ball will slow down to the same height $$h$$, since no energy is assumed lost here, and then fall back.

In this way the ball will "oscillate", but it is not really harmonic (the general solution won't be a simple sine wave).

Your difficulty results because you are confounding simple harmonic motion and periodic motion.

In SHM, the force is indeed linear in the displacement and towards the equilibrium position. Basically: $$F=-kx$$. Alternatively the potential energy is $$U=\frac12 kx^2$$. One feature of SHM is that the period is independent of the amplitude of the motion.

Ordinary periodic motion only requires that $$x(t+T)=x(t)$$ for some period $$T$$. Any confining potential in 1d will produce periodic motion since the particle will repeatedly go between the two turning points of the motion. Of course, computing the period $$T$$ for such a system may be difficult and will usually depend on the amplitude of the motion (as in your example) but it is still periodic motion.

The simplest example of periodic but not SHM would be a pendulum with an initial angle $$\theta_0$$ that is not small. Certainly the motion is periodic but for large angle the period has a complicated expression that depends on $$\theta_0$$.