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Chapter 5: Problem 26

A hockey stick stores \(4.2 \mathrm{~J}\) of potential energy when it is bent \(3.1 \mathrm{~cm}\). Treating the hockey stick as a spring, what is its spring constant?

Short Answer

Expert verified
The spring constant is approximately \( 8742.86 \mathrm{~N/m} \).

Step by step solution

01

Understand the Formula for Spring Potential Energy

The potential energy stored in a spring is given by the formula \( PE = \frac{1}{2} k x^2 \), where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its equilibrium position.
02

Substitute Given Values into the Formula

We know that \( PE = 4.2 \mathrm{~J} \) and \( x = 3.1 \mathrm{~cm} = 0.031 \mathrm{~m} \). Substitute these values into the formula to get \( 4.2 = \frac{1}{2} k (0.031)^2 \).
03

Solve for the Spring Constant

Rearrange the equation to solve for \( k \): \( k = \frac{2 \times 4.2}{(0.031)^2} \).
04

Calculate the Result

Perform the calculation: \( k = \frac{8.4}{0.000961} \approx 8742.86 \). This results in a spring constant of approximately \( 8742.86 \mathrm{~N/m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spring Potential Energy
When we talk about spring potential energy, we refer to the energy stored in an object due to its position or configuration. For springs, this energy is specifically associated with how much the spring is stretched or compressed. Spring potential energy is calculated using the formula:\[ PE = \frac{1}{2} k x^2 \]where:
  • PE stands for potential energy.
  • k is the spring constant, which indicates the stiffness of the spring.
  • x represents the displacement, or how far the spring is stretched or compressed from its equilibrium position.
This formula highlights the relationship between the spring constant and the displacement. The stronger the spring, the higher the spring constant, meaning it can store more energy.
Displacement
Displacement in the context of a spring refers to the distance that the spring is stretched or compressed from its natural position, known as the equilibrium position. Displacement is an essential variable in calculating spring potential energy because it demonstrates how much energy is stored in the spring when it is moved from its rest position. Additionally, displacement is expressed in meters (m) in the SI unit system. For instance, in our exercise, the hockey stick was bent for a displacement of 3.1 cm, which is converted to meters as 0.031 m for the formula usage. This small detail ensures the units remain consistent, leading to accurate calculations.
Equilibrium Position
The equilibrium position is a vital concept in understanding the behavior of springs. It refers to the state where a spring is neither compressed nor stretched - essentially, it's at rest. At this position, there is no net force acting on the spring. When an external force is applied to stretch or compress the spring, the spring tries to return to its equilibrium position due to its elasticity. This action is where potential energy comes into play, as the energy is stored based on how far the spring is displaced from this natural state. Understanding the equilibrium position is critical when examining oscillatory motions and forces. By calculating how far a spring shifts from this position (displacement), we can determine the energy stored and ascertain the spring constant to assess the spring's behavior.

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