So a pendulum clock designed to keep time with small oscillations of the pendulum will gain four seconds an hour or so if the pendulum is made to swing with a maximum angular displacement of ten degrees. The total energy is therefore. The analysis of pendulum motion in terms of angular displacement works for any rigid body swinging back and forth about a horizontal axis under gravity.
For example, consider a rigid rod. Near that minimum, the height function and its second derivative can be exactly matched by a circle, and the first derivative vanishes, so you can get approximation accurate up to a 3rd-order error. If the curve is left-right symmetric about the bottom point, then the error shrinks to 4th order.
Show 3 more comments. Alfred Centauri Alfred Centauri The answers including the accepted answer that state the physical intuition is that there is higher potential energy at max displacement for larger amplitudes and thus higher kinetic energy at max velocity are wrong on my view; this intuition does not imply that the amplitude and period are independent. He wanted "physical intuition" on why this works.
This answer is entirely related to the differential equations, not the physics. Although I agree there are plenty of nuances to this question, i believe he accepted my answer because it was a simple physical explanation for why they can have the same period. I purposefully avoided math and merely provided intuition on how it was possible from a physics school perspective. It seemed to be what the OP wanted more intuition about. Also, it isn't true that my answer is entirely related to differential equations; the concepts of linearity and superposition are not limited to the context of differential equations.
Indeed, I could remove the phrase "differential equation" from my answer as it isn't actually required. Show 7 more comments. Enns M. Enns 6, 4 4 gold badges 23 23 silver badges 36 36 bronze badges.
I said that the scale invariance of periods implies linear differential equations' periodic solutions have amplitude-independent periods the result to be explained. I thus provided an explanation in terms of another fact. David I. McIntosh David I. McIntosh 1 1 bronze badge. In fact, any convex potential function has the property that the acceleration gets ever larger as the amplitude increases, but most of these potentials do not give rise to isochronous orbits , which is what the question is about.
In particular the standard circular pendulum is not isochronous, it only becomes approximately isochronous in the low-amplitude limit. One could continue but the key feature is already there.
If the amplitude doubled so would the distance covered in a given time. Thus the period of the motion does not depend on the amplitude. For a large initial angle, the difference between the small angle approximation black and the exact solution light gray becomes apparent almost immediately. Oscillation of a Simple Pendulum The Equation of Motion A simple pendulum consists of a ball point-mass m hanging from a massless string of length L and fixed at a pivot point P.
When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. With the assumption of small angles, the frequency and period of the pendulum are independent of the initial angular displacement amplitude.
A pendulum will have the same period regardless of its initial angle. What is amplitude and frequency? Is frequency directly proportional to amplitude? What happens to energy when amplitude increases? What is amplitude frequency and time period?
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