Question

How does the potential energy of a spring change if its amount of stretch is doubled?c

Short answer

The potential energy of the spring quadruples when the stretch is doubled.

Step-by-step solution

Understand the Formula for Potential Energy of a Spring

The potential energy stored in a spring is given by the formula \( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement or stretch of the spring from its equilibrium position.

Analyze the Effect of Doubling the Stretch

If the stretch \( x \) is doubled, it becomes \( 2x \). Substitute \( 2x \) into the potential energy formula. The new potential energy becomes \( PE' = \frac{1}{2} k (2x)^2 \).

Simplify the Expression

Simplify \( PE' = \frac{1}{2} k (2x)^2 \), which equals \( PE' = \frac{1}{2} k (4x^2) \). This simplifies further to \( PE' = 2kx^2 \).

Compare the New Potential Energy with the Original

The original potential energy was \( PE = \frac{1}{2} k x^2 \). The new potential energy \( PE' = 2kx^2 \) is four times the original potential energy \( PE' = 4 \times \frac{1}{2} k x^2 \). Hence, the potential energy quadruples.

Key concepts

Spring Constant

The spring constant, often denoted as \( k \), is a key value in understanding how springs behave under force. It tells you how stiff or flexible a spring is. A high spring constant means the spring is very stiff and requires a lot of force to stretch or compress it by a unit distance. Conversely, a low spring constant indicates a more flexible spring that is easier to stretch.Several factors influence the spring constant:

• Material of the spring: Springs made from stiffer materials have higher spring constants.
• Coil thickness: Thicker coils in a spring generally lead to a higher spring constant.
• Number of coils: A spring with more coils will have a different constant compared to one with fewer coils.

Understanding the spring constant is crucial when calculating the potential energy of a spring because it directly affects how much energy is stored as the spring is compressed or extended.

Hooke's Law

Hooke's Law is a principle that describes how much force is needed to stretch or compress a spring by a certain amount. It is represented by the formula \( F = -kx \), where:

• \( F \) is the force applied to the spring.
• \( k \) is the spring constant, indicating the spring's stiffness.
• \( x \) is the displacement from the spring's equilibrium position.

This law tells us that the force needed to compress or extend a spring is directly proportional to the distance it is stretched or compressed. The negative sign in the equation indicates that the force exerted by the spring is in the opposite direction to the displacement.
Hooke's Law is foundational in physics as it helps in understanding not only springs but also any elastic material which follows similar properties. It allows us to calculate the force involved in various mechanical systems and plays a significant role in mechanical energy calculations.

Mechanical Energy

Mechanical energy is the sum of potential energy and kinetic energy in a system. In the context of springs, mechanical energy primarily involves the potential energy stored when the spring is compressed or stretched.The potential energy stored in a spring is calculated using the formula:\[ PE = \frac{1}{2} k x^2 \]Where:

• \( k \) is the spring constant.
• \( x \) is the displacement from the spring's natural position.

The potential energy depends on both the spring constant and the displacement—the more you compress or stretch a spring, the more energy it stores.
Mechanical energy conservation in springs implies that, in the absence of friction and external forces, the energy stored as potential energy will convert into kinetic energy and vice versa. This allows for the prediction of motion in systems like pendulums, trampolines, or shocks in car suspension where springs play a crucial role.