- #1
Kaushik
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- TL;DR Summary
- ##x(t) = Acos(2πt/T)##
Why do we include ##2π/T## and not just ##t## in equation of simple harmonic oscillators?
##x(t) = Acos(2πt/T)##
##x(t) = Acos(2πt/T)##
A simple answer is that there are 2π radians in a circle and t/T represents a fraction of a complete cycle. The wave repeats every T (the period). Then what @Doc Al said.Kaushik said:Summary:: ##x(t) = Acos(2πt/T)##
Why do we include 2π/T
The inclusion of 2π/T in mathematical equations is due to the use of radians as a unit of measurement for angles. Radians are a more natural unit for measuring angles in many situations, and using 2π/T allows for easier conversion between radians and other units of measurement, such as degrees.
In physics and mathematics, frequency (f) and period (T) are inversely related. The frequency is the number of cycles per unit time, while the period is the time it takes for one complete cycle. Including 2π/T in equations allows for the conversion between these two quantities.
Trigonometric functions, such as sine and cosine, are used to describe the relationship between the sides and angles of a right triangle. These functions are periodic, meaning they repeat themselves after a certain interval. The value 2π/T is used to represent one full cycle of a trigonometric function, making it a useful tool in solving equations and graphing these functions.
In physics, many natural phenomena, such as waves and oscillations, can be described using sine and cosine functions. These functions have a period of 2π/T, which is why this value is commonly used in physics equations. It allows for the description and prediction of these phenomena in a mathematical and scientific way.
Yes, the value 2π/T is used in various fields, including engineering, economics, and computer science. It is a fundamental concept in mathematics and has applications in many other disciplines, making it a crucial tool for scientists and researchers.