You're riding in a friend's \(1400-\mathrm{kg}\) car with bad shock absorbers, bouncing down the highway at \(20 \mathrm{m} / \mathrm{s}\) and executing vertical SHM with amplitude \(18 \mathrm{cm}\) and frequency \(0.67 \mathrm{Hz}\) Concerned about fuel efficiency, your friend wonders what percentage of the car's kinetic energy is tied up in this oscillation. Make an estimate, neglecting the wheels' rotational energy and the fact that not all of the car's mass participates in the oscillation.

Simple harmonic motion (SHM) is a type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the opposite direction. It's characterized by its sinusoidal nature, which means it follows a repetitive path in a smooth, wave-like pattern. Imagine a child swinging steadily back and forth on a playground swing; that motion approximates simple harmonic motion.

When we look at the scenario of the car with bad shock absorbers, the vertical bouncing of the car is an example of SHM. The car moves up and down about an equilibrium position and the shocks provide the restoring force that creates the SHM pattern. To describe SHM, we focus on key parameters like amplitude, frequency, and kinetic energy. The amplitude indicates how far from the equilibrium position the object travels (in this case, 18 cm), while the frequency (0.67 Hz) tells us how often the complete oscillation occurs per second.

In physics, kinetic energy refers to the energy that an object possesses due to its motion. For a body in linear motion, it's calculated using the formula: \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.

In the context of simple harmonic motion, the velocity and thus the kinetic energy varies throughout the oscillation. The maximum kinetic energy is found when the object passes through the equilibrium point because this is where it achieves its maximum speed. To calculate the maximum kinetic energy (\( K_{max} \)) of an object undergoing SHM around the vertical axis, we apply a slightly different formula: \( K_{max} = \frac{1}{2} m A^2 (2 \pi f)^2 \), where \( A \) is the amplitude and \( f \) is the frequency of the oscillation. This equation takes into account that the kinetic energy is maximized at the largest velocity, which occurs as the object crosses the equilibrium point.

Oscillation frequency is a fundamental concept when discussing any periodic motion, including simple harmonic motion. It refers to the number of oscillations or cycles that occur in one second, measured in Hertz (Hz). The frequency tells us about the rapidity of the oscillation. In the given problem, the car's oscillation frequency is 0.67 Hz, indicating that the car completes approximately two-thirds of a cycle every second.

The frequency is an essential factor in determining kinetic energy in SHM since it affects the velocity of the oscillating object. The relationship between kinetic energy and frequency is squared in the equation \( K_{max} = \frac{1}{2} m A^2 (2 \pi f)^2 \), which means a small increase in frequency can lead to a significant increase in the kinetic energy of the object. Understanding the oscillation frequency is pivotal for interpreting the behavior of the system and for calculating the energy involved in the motion.