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11th Physics Online Notes -The Simple Pendulum

11th Physics Online Notes -The Simple Pendulum 

Class 11 Physics: A simple pendulum is just a pendulum made out of a small object attached to a light string. Ideally, it is a “point” particle attached to a massless string which is fixed to a pivot point. If the pendulum is displaced from equilibrium, it swings back and forth, and its motion is periodic. The questions we want to consider are: is the motion simple harmonic, and what is the equation for the period T of the motion?
To answer these questions, one starts with the equation relating forces and motion. I am going to use a different variable than the textbook. I will specify the position of the particle by the distance, along an arc, that the particle is from the equilibrium position. See the figure at the end of this section. If the length of the pendulum is l, the ratio x/l is the angle θ (in radians) that the string makes with the vertical. Most textbooks use the angle θ to specify the location of the particle, so x = lθ is the connection between the text’s θ and our x. I think this will be easier for us to make reference to the spring equation and avoid using torque.
When the pendulum is hanging vertical, x = 0, right displacement is positive x and left displacement is negative x. If an object is displaced a distance x along the arc, the component of gravity in the direction of the arc is −mgsin(θ) or −mgsin(x/l). The negative sign means that the force of gravity is a restoring force. If x > 0, sin(x/l) > 0 and the force is in the negative direction. If x < 0, sin(x/l) < 0 and the force is in the positive direction. Thus,
Fx = −mgsin( x/l )

For a force to produce simple harmonic motion, the force must be proportional to −x. The equation for the simple pendulum is NOT of this form. Here the force is proportional to −sin(x/l). Thus, the motion of a simple pendulum is NOT simple harmonic. Don’t get confused with the “sin” function. A sinusoidal restoring force does not produce perfect sinusoidal motion. Only a linear restoring force gives perfect sinusoidal motion. This is easy to demonstrate. For simple harmonic motion, the period does not depend on the amplitude. Pendula with a larger amplitude (larger xmax) have a longer period than ones with a smaller amplitude (smaller xmax).
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