Energy Analysis in Simple Harmonic Motion
An oscillator executing simple harmonic motion is described by a mass \(m = 0.50 \text{ kg}\), angular frequency \(\omega = 2.0 \text{ rad s}^{-1}\) and amplitude \(x_0 = 0.10 \text{ m}\). Using the energy relations \(E_T = \tfrac{1}{2} m \omega^2 x_0^2\) and \(E_p = \tfrac{1}{2} m \omega^2 x^2\), answer the following: (a) Calculate the total […]
Calculating the Period of a Mass-Spring System
A mass-spring system is set up on a frictionless horizontal surface. A mass of \( m = 0.50 \text{ kg} \) is attached to a spring with a spring constant of \( k = 200 \text{ N m}^{-1} \). Using the formula \( T = 2\pi\sqrt{m/k} \), calculate the period of oscillation. Show all your […]
Maclaurin Series Application in Small-Angle Approximations
Consider the function \(sin(x)\) and its Maclaurin series expansion. In many physical applications, particularly in simple harmonic motion, the small-angle approximation \(sin(\theta) \approx \theta\) is used to simplify the equations of motion. (a) Using the Maclaurin series, expand \(sin(x)\) up to and including the cubic term. (b) Explain how the small-angle approximation derived from this […]