Question

Should you use radians or degrees when solving SHM problems?

Answer 1

When solving Simple Harmonic Motion (SHM) problems, it is generally advisable to use radians rather than degrees. The mathematical equations governing SHM, such as the sine and cosine functions, are more naturally expressed in radians. Using radians simplifies the equations and aligns with the standard mathematical conventions in SHM analysis.

Step-by-step explanation:

Simple Harmonic Motion involves oscillatory motion that can be described by trigonometric functions like sine and cosine. The fundamental formula for SHM is
`\(x(t) = A \sin(\omega t + \phi)\)`, where A is the amplitude,
`\(\omega\)` is the angular frequency, t is time, and
`\(\phi\)` is the phase angle. In this formula,
`\(\omega\)` is given in radians per unit time. Using radians is advantageous because the argument of trigonometric functions is naturally expressed in radians, making the mathematical representation of SHM simpler and more coherent.

For instance, if you have a problem with an angular frequency of
`\(2\pi\)` radians per second and want to find the displacement at seconds, the use of radians allows for a direct substitution into the formula:
`\(x\left((\pi)/(2)\right) = A \sin\left(2\pi \cdot (\pi)/(2) + \phi\right)\)`. This simplifies calculations and aligns with the standard practice in physics and engineering where angular quantities are commonly measured in radians.

In summary, using radians when solving SHM problems facilitates simpler mathematical expressions and aligns with the natural representation of angular quantities in trigonometric functions, making the analysis more intuitive and consistent with standard mathematical conventions.