Question

A spring is stretched \(5.00 \mathrm{~cm}\) from its equilibrium position. If this stretching requires \(30.0 \mathrm{~J}\) of work, what is the spring constant?

Short answer

Answer: The spring constant is 24000 N/m.

Step-by-step solution

Identify the formula for work done on a spring

The formula for work done on a spring is given by: $$W = \frac{1}{2}kx^2$$ where \(W\) is the work done, \(k\) is the spring constant, and \(x\) is the distance stretched from equilibrium.

Simplify the formula for spring constant

We need to find the spring constant \(k\). We can simplify the formula from Step 1 to solve for \(k\). So, $$k = \frac{2W}{x^2}$$

Plug in the given values

We are given \(W = 30.0 \mathrm{~J}\) and \(x = 5.00 \mathrm{~cm}\) which is equal to \(0.05 \mathrm{~m}\) (converted to meters). Now, we can input these values into the formula we derived in Step 2: $$k = \frac{2\times30.0}{(0.05)^2}$$

Calculate the spring constant

Finally, let's calculate the spring constant \(k\): $$k = \frac{60}{0.0025} = 24000 \mathrm{~N/m}$$ The spring constant is \(24000 \mathrm{~N/m}\).

Key concepts

Work Done on a Spring

When we talk about the work done on a spring, we are referring to the energy needed to stretch or compress it from its natural length (also known as its equilibrium position). In physics, work is defined as a force causing the displacement of an object. In the context of a spring, that force is directly related to the spring's displacement from equilibrium.

To calculate this, we use the formula \( W = \frac{1}{2}kx^2 \), where \(W\) is the work done on the spring (in joules), \(k\) is the spring constant (in newtons per meter), and \(x\) is the displacement from the equilibrium position (in meters). The importance of this concept lies in energy conservation; the work done on the spring turns into potential energy, which can later be converted into other forms of energy, such as kinetic energy—playing a crucial role in understanding mechanical systems.

Hooke's Law

Moving onto Hooke's law, it's a fundamental principle in physics that tells us how much force is needed to extend or compress a spring by a certain amount. According to this law, the force \( F \) required to stretch or compress a spring is directly proportional to the displacement \( x \)—the distance it's stretched or compressed from its original length. Mathematically, this is expressed as \( F = -kx \). The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement, meaning the spring naturally acts to return to its equilibrium.

Hooke's law is essential for solving problems involving springs because it enables us to determine the spring constant \( k \), which gives us a measure of the spring's stiffness. In the textbook exercise, we utilized Hooke's law to understand the relationship between the work done and the spring constant.

Physics Problem Solving

Approaching physics problem solving requires a systematic method to untangle the often complex scenarios we encounter. In our spring constant problem, the process began with identifying the relevant formula, which in this case was related to the work done on a spring. The next step involved isolating the desired variable—here, the spring constant \( k \).

By rearranging the formula and substituting the given quantities, we incrementally worked our way toward the solution. This methodical approach—starting from known physical laws, applying them to the given values, and manipulating the equations accordingly—is at the heart of problem solving in physics. Encouraging the use of unit conversions, symbol definitions, and algebraic manipulations helps demystify complex problems and builds foundational skills for students.

Mechanical Energy

Finally, let's discuss mechanical energy. Mechanical energy is the sum of potential energy and kinetic energy present in the components of a mechanical system. For a spring, the work done to stretch or compress it is stored as potential energy within the spring, often referred to as elastic potential energy due to the nature of the spring's deformation.

Understanding mechanical energy is crucial as it allows for the analysis of energy transfer within a system. For example, when a compressed spring is released, the stored potential energy is converted to kinetic energy, which could set an object in motion. The conservation of mechanical energy principle enables us to predict the behavior of physical systems over time. In the textbook problem, by calculating the work done on the spring, we indirectly determined the potential mechanical energy stored in the spring at a particular displacement.