91Ó°ÊÓ

Hooke's Law is a fundamental principle in physics that explains how springs work. It states that the force needed to extend or compress a spring is directly proportional to the distance it is stretched or compressed. This relationship can be written as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position.

This principle helps us understand why a spring stretches when a weight is hung from it. In the original exercise with the apple, the spring stretches due to the weight of the apple it holds, consistent with Hooke's Law. By using the equation \( x = \frac{F}{k} \), we can calculate that the apple, exerting a force of 1.00 N, stretches the spring by 0.67 m. This understanding is crucial for solving problems that involve spring displacement due to weight or force.

A pendulum is a simple mechanical system consisting of a mass (called a bob) hanging from a pivot. It swings back and forth under the influence of gravity. When the pendulum swings through small angles, its motion closely resembles that of simple harmonic motion (SHM). This similarity makes it useful for exploring concepts in physics like periodic motion and frequency.

For the apple in the exercise, when it stops bouncing and swings side to side, it moves like a simple pendulum. In such a setup, its frequency of swinging is determined by the length of the pendulum and acceleration due to gravity. Interestingly, the frequency of a pendulum is half of the bouncing frequency from the spring motion, as specified in the exercise. Understanding this relationship is important for calculating the pendulum's effective length.

The spring constant, denoted by \( k \), describes the stiffness of a spring. It tells us how resistant a spring is to being compressed or stretched. The larger the spring constant, the stiffer the spring, and more force is required to stretch or compress it by a given amount.

A critical part of the exercise is knowing the spring constant to predict how much the spring will stretch under a specific force. Here, with a spring constant of 1.50 N/m, the spring stretches 0.67 m when a 1.00 N force from the apple is applied. For any given spring system, knowing \( k \) allows us to calculate forces and displacements accurately, and is essential to problems dealing with springs.

Frequency calculation is part of understanding oscillatory motions, such as those seen in springs and pendulums. It tells us how many oscillations occur in a given time period, normally interpreted as hertz (Hz), which means cycles per second.

In the context of the exercise, two types of frequencies are crucial: the frequency of the spring bouncing (SHM) and the pendulum swinging frequency. To find the spring system's frequency, the exercise uses the formula \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \). After identifying that the mass of the apple is approximately 0.102 kg, it is plugged into this formula along with the spring constant to determine the frequency of 0.61 Hz for the bouncing spring.

For the pendulum, the frequency is half of the spring frequency, calculated to be around 0.305 Hz, leveraging its physical pendulum formula \( f_p = \frac{1}{2\pi} \sqrt{\frac{g}{L}} \). Both calculations demonstrate important applications of frequency determination for different oscillating systems.