Question

A force of \(2 \mathrm{~N}\) stretches a spring \(5 \mathrm{~cm}\). How much work is performed in stretching the spring?

Short answer

The work performed in stretching the spring is 0.05 Joules.

Step-by-step solution

Understand Hooke’s Law

Hooke’s Law states that the force needed to stretch or compress a spring by some distance is proportional to that distance. Mathematically, this is expressed as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the distance the spring is stretched or compressed.

Identify Known Quantities

From the problem, we know the force \( F = 2 \mathrm{~N} \) and the distance \( x = 5 \mathrm{~cm} = 0.05 \mathrm{~m} \).

Calculate the Spring Constant \( k \)

Using Hooke's Law formula \( F = kx \), solve for \( k \): \[ k = \frac{F}{x} = \frac{2 ~\mathrm{N}}{0.05 ~\mathrm{m}} = 40 ~\mathrm{N/m} \]

Use the Work Formula for Springs

The formula for work done in stretching or compressing a spring is \( W = \frac{1}{2} k x^2 \).

Calculate the Work Done

Substitute the values for \( k \) and \( x \) into the work formula:\[ W = \frac{1}{2} \times 40 ~\mathrm{N/m} \times (0.05 ~\mathrm{m})^2 = \frac{1}{2} \times 40 \times 0.0025 = 0.05 ~\mathrm{J} \]

Key concepts

Hooke's Law

Hooke's Law is a fundamental principle in the study of elasticity, particularly in springs. It establishes a relationship between the force applied to a spring and the distance the spring stretches. This relationship is expressed mathematically as:\[ F = kx \]- **F**: the force applied to the spring (in newtons, N).- **k**: the spring constant (in N/m), a measure of the spring's stiffness.- **x**: the displacement (in meters, m) from the spring's original position.
This means the more you stretch or compress a spring, the greater the force required. Hooke's Law is applicable only within the elastic limit, i.e., the range where the spring behaves elastically and returns to its original shape after the force is removed.

Spring Constant

The spring constant, denoted as **k**, is a measure of a spring's stiffness. This constant is unique to each spring and determines how much force is needed to stretch or compress the spring by a certain distance.
To find the spring constant, rearrange Hooke's Law as follows:\[ k = \frac{F}{x} \]
- **F** is the force applied (N).- **x** is the displacement (m).
This formula shows that a larger spring constant means a stiffer spring, which requires more force to stretch or compress it. In the given exercise, the spring constant was calculated as 40 N/m, meaning it takes 40 newtons to stretch the spring by one meter.

Force and Distance

Force and distance are crucial components in calculating work done on a spring. The force is the strength required to change the spring's length, while the distance is how much the spring stretches or compresses from its initial position.
When a force is exerted on a spring, it displaces the spring by a distance **x**. The relationship between force and distance in a spring follows Hooke’s Law, which is linear under normal conditions:
- A bigger force results in a larger displacement (assuming the spring does not reach its elastic limit).
In the scenario provided, a 2-N force moves the spring 5 cm (0.05 m), demonstrating this direct proportionality.

Work Done

Work done on a spring is a measure of energy transferred when a force stretches or compresses the spring over a distance. Calculating the work done requires understanding the force-distance relationship and using the formula for springs:
\[ W = \frac{1}{2} k x^2 \]
- **W**: work done (joules, J).- **k**: the spring constant (N/m).- **x**: displacement (m).
This equation accounts for the fact that not all applied force does consistent work over the stretch distance (due to changing force as the spring stretches). The result is the total energy needed to move the spring from its initial position. For instance, with a spring constant of 40 N/m, stretching it by 0.05 m requires 0.05 joules of work. This calculation gives us insights into the energy dynamics when dealing with springs.