Work Done on a Spring

Unlike work done by a constant force ($W = Fd$), the force required to stretch a spring increases linearly with extension according to Hooke's Law: $F = ke$.

Because the force is variable, the total work done is calculated as the average force multiplied by the total extension, which leads to the factor of $\frac{1}{2}$ in the energy formula.

Geometrically, the work done is equal to the area under the Force-Extension graph. Since the relationship is linear, this area forms a triangle with base $e$ and height $F$, resulting in $\text{Area} = \frac{1}{2} \times e \times F$.

Substituting $F = ke$ into the area formula yields the standard energy equation: $E_e = \frac{1}{2} k e^2$

Step 1: Identify the Spring Constant: Determine the value of $k$ in N/m. If the gradient of a Force-Extension graph is provided, that gradient represents $k$.

Step 2: Calculate Extension: Ensure you are using the extension ($e$), not the total length. Subtract the original length from the final length and convert to meters (m).

Step 3: Apply the Formula: Substitute the values into $E_e = \frac{1}{2} k e^2$. Remember that only the extension is squared, not the entire expression.

Step 4: Verify Units: The resulting energy will be in Joules (J) only if $k$ is in N/m and $e$ is in meters.

Extension vs. Length: Extension is the change in length. Using the total length of a spring in the formula is a common error that leads to significantly inflated energy values.

Compression vs. Stretching: The formula $E_e = \frac{1}{2} k e^2$ applies equally to both compression and stretching, as the square of the displacement ($e^2$) will always be positive.

The Square Rule: Always check if you squared the extension. If the extension doubles, the work done increases by a factor of four ($2^2 = 4$).

Unit Conversion Check: Exams often provide extension in centimeters (cm) or millimeters (mm). Always convert these to meters (divide by 100 or 1000) before calculating energy.

Graph Interpretation: If asked to find work from a graph, calculate the area of the triangle. Do not simply multiply the final force by the final extension, as this would calculate a rectangle (double the actual work).

Limit of Proportionality: Look for keywords like 'elastically' or 'Hooke's Law applies'. If the spring has exceeded its limit, the standard formula cannot be used.

Linear Force Assumption: Students often mistakenly use $W = Fd$ with the final force. This is incorrect because the force starts at zero and increases; the work is the average effect of this variable force.

Squaring the Product: Ensure you calculate $\frac{1}{2} \times k \times (e^2)$ rather than $(\frac{1}{2} \times k \times e)^2$. The exponent applies only to the extension variable.

Negative Extension: In compression, students sometimes try to use a negative value for $e$. While $e$ is technically a displacement vector, the energy stored is a scalar and depends on $e^2$, making the sign irrelevant for the final energy value.