Pendulum

Syntax

Overview

Pendulums fall into a class of simple harmonic oscillators in which the elasticity is due to gravity rather than that of a spring or twisted wire being compressed or stretched. The Simple Pendulum consists of a mass suspended at the end of a massless string that is fixed at one end. A Physical Pendulum can have a complicated distribution of mass suspended at the end of a massless string and the movements don't necessarily have to move from left to right.

Equations

Torque = -Length (L)(Force of gravity (F_gsin(theta))
Inertia (I) = (1/12)ML²
Period (T) = 2pi (sqrt(L / g) = 2pi(sqrt(I/mgh))
-L(mgsin(theta)) = I*(alpha)
Alpha = -(mgh / I) * theta
Velocity (v) = -wx_msin(wt + phase constant or phase angle (phi))
Acceleration (a) = -w²x_mcos(wt + phi)

Question 1

A physical pendulum consists of a meter stick that is pivoted at a small hole drilled through a stick a distance d from the 50 cm mark. The period of oscillation is 2.5s find d.

Answer

Since this is a physical pendulum, T = 2pi(sqrt(I/mgh)). The rotational inertia is calculated using I=(1/12)ML²
therefore, T=2pi(sqrt((ML²/12)+md²/mgd) = 2pi(sqrt((L²/12gd) + (g/d))
then, d=(g(T/2pi²)+-(sqrt(d²(T/2pi)⁴ - (L²/3))))/2
plugging in the values gives and choosing the plus sign gives you an impossible value for d, therefore choose the minus sign, which gives you 0.056m for d, which, if converted to centimeters, equals 5.6cm

Question 2

A pendulum is formed by pivoting a long string of length L, that is massless about a point on the string that is a distance d above the center of the string. (a) Find the period of this pendulum in terms of d, L, and g, assuming small-amplitude swinging. What happens to the period if (b) d is decreased, or (c) L is increased

Answer

(a)T=2pi(sqrt(((L/2)+d)/g)
(b)As d increases, the period increases and as d decreases, the period increases.
(c)As L increases, the period increases and as L decreases, the period increases.

Special Cases

A Special Case would be the ballistic pendulum. The ballistic pendulum consists of a mass with a velocity v providing a force to a mass suspended at the end of a string through elastic collision.
For an example, follow this link:
http://farside.ph.utexas.edu/teaching/301/lectures/node96.html

Applications

Clocks

Use the pendulum to regulate the motion of the hands of a clock, such as in grandfather clocks. Everytime the pendulum swings the gear moves one notch and this leads to other gears moving the hands of the clock

Foucault Pendulum

This is a pedulum that demonstartes the Earth's Rotation. This is a very long pendulum. being several stories high. The reasoning behind this is because the longer length will cause it to swing for a longer time. To people viewing this pendulum, The pendulum seems to change directions, but the Pendulum is not changing direction but the Earth is moving beneath it. There are many of these, but the most famous one is at the Pantheon in Paris, France.

External Links

• [http://hyperphysics.phy-astr.gsu.edu/Hbase/balpen.html] - Further Explanation
• [http://www.school-for-champions.com/science/pendulum.htm] - Further Explanation
• [http://webphysics.davidson.edu/physlet_resources/bu_semester1/c12_ballistic.html] - This has a demonstration that you can watch

Joke