-1

It is known that in simple harmonic motion, the total energy of the system is proportional the square of the amplitude, but how can I prove that for a simple pendulum where amplitude is the arc length of the part of the circle? Much obliged for any help.

cwwwonggg
  • 23
  • 1
    How can amplitude be equal to arc length? – Courage Nov 14 '15 at 10:56
  • @Vishwaas then what is the amplitude? Not the length that the mass move from the equilibrium position to the extreme point? – cwwwonggg Nov 14 '15 at 11:00
  • Are you working with pendulum undergoing SHM? Then the arc can be considered as a straight line rather than a curve. Otherwise, it doesn't go SHM. –  Nov 14 '15 at 11:12

1 Answers1

1

Suppose the maximum angle of the pendulum to the vertical direction is $\theta$, where the pendulum has a zero-velocity. According to the conservation of mechanical energy, the total energy of the system, $E$, should be:

$$E=mgl(1-\cos\theta )$$
If $\theta$ is small enough, we can expand $\cos\theta$ up to the second order:

$$\cos(\theta)\approx 1-\frac{\theta^2}{2}$$

Plug this into the energy formula, we can get:

$$E=mgl(1-\cos(\theta))\approx mgl\left[1-\left(1-\frac{\theta^{2}}{2}\right)\right]=\frac{mgl\theta^2}{2}$$

The total energy of the simple pendulum system is proportional to $\theta^2$. It should be noted that this relation just holds when $\theta$ is very small, only based on which $1-\frac{\theta^2}{2}$ can be a good approximation of $\cos\theta\; .$

bitsoal
  • 54