Hooke's Law and Elastic Potential Energy

This paragraph introduces Hooke's Law, a fundamental principle in physics that describes the relationship between the force applied to a spring and the displacement it causes. It explains how a spring's force (Fp) to stretch it is directly proportional to the displacement (x), expressed as Fp = kx, where k is the spring constant. The concept of the elastic region is introduced, highlighting that the proportionality between force and displacement holds up to a certain limit. The paragraph also explains the concept of the restoring force (Fs or Fr), which acts in the opposite direction to return the spring to its original length. The equilibrium state of the spring is discussed, where the applied force and the restoring force are equal in magnitude but opposite in direction.

This paragraph focuses on the practical application of Hooke's Law to calculate the spring constant (k) using the formula Fp = kx. A specific example is provided where a force of 200 newtons stretches the spring by 4 meters, and the calculation shows that the spring constant is 50 newtons per meter. The paragraph further explains the implications of the spring constant, indicating that a higher value signifies a stiffer spring, which requires more force to stretch by a given distance. The concept is illustrated by comparing two springs with different spring constants and explaining how the stiffness varies.

This paragraph continues the exploration of Hooke's Law by demonstrating how to calculate the force required to stretch a spring by a given distance, using the provided spring constant (k). The example uses a spring constant of 300 newtons per meter and calculates the force needed to stretch the spring by 45 centimeters, resulting in 135 newtons. The process involves converting the displacement from centimeters to meters and applying the formula. The paragraph also emphasizes the importance of checking the reasonableness of the answer by comparing it to other related quantities.

This paragraph presents two methods for calculating the stretch distance of a spring when a specific force is applied. The first method involves finding the spring constant (k) from one force-displacement pair and then using it to calculate the stretch distance for a different force. The second method uses the ratio of forces to displacements, which, due to the constant k, can be directly applied to find the unknown displacement. The paragraph then introduces the concept of work done in stretching a spring, explaining that the work is equal to half the product of the spring constant and the square of the displacement (delta x). The relationship between work and elastic potential energy is also discussed, noting that the work done is positive when the force and displacement are in the same direction.

This paragraph delves into the concept of elastic potential energy stored in a spring when it is stretched or compressed. It explains that the work done to stretch a spring is equal to the change in elastic potential energy. The formula for elastic potential energy is derived as one half times the spring constant times the square of the displacement from the natural length (k * x^2). The paragraph provides an example calculation for the elastic potential energy stored when a spring is stretched by 25 centimeters with a 450 newton force, resulting in 56.25 joules of energy. It also clarifies the difference between the work done (which depends on the change in position) and the potential energy (which depends on the position relative to the natural length of the spring).

This paragraph discusses the work required to stretch a spring from one position to another, given a spring constant. It emphasizes the importance of calculating the change in potential energy, which is equal to the work done if the net acceleration is zero. The paragraph provides a detailed example where the spring is stretched from 4.5 centimeters to 8.2 centimeters with a spring constant of 150 newtons per meter. The work required is calculated as 0.352 joules. The paragraph also explains the difference between positive and negative work, depending on the direction of the force relative to the displacement. It concludes by reiterating that the work done by the pulling force is positive, while the work done by the spring's restoring force is negative.