Question

A spring has a force constant of \(15.0 \mathrm{~N} / \mathrm{cm} .(a)\) How much work is required to extend the spring \(7.60 \mathrm{~mm}\) from its relaxed position? ( \(b\) ) How much work is needed to extend the spring an additional \(7.60 \mathrm{~mm}\) ?

Short answer

The work required to extend the spring first 7.60 mm is \(W_1\) and the work required for another 7.60 mm is \(W_2 = W_{total} - W_1\) where the values can be obtained from the above calculations.

Step-by-step solution

Calculate work done for first extension

Use the formula \(W = 1/2 * k * X^2\) where k is the spring constant and X is the extension. Substituting the given values, we get \(W = 1/2 * 15.0 N/cm * (7.60 mm)^2).\ Convert 7.60mm to cm to match the spring constant's units, then perform the operation.

Calculate work done for second extension

The total work done to stretch the spring by 2* 7.60 mm would be \(W_{total} = 1/2 * k * (2*X)^2\). Compute this value. The work done for the second extension is the difference between the \(W_{total}\) and the work done for the first extension. Thus, subtract the first work value from the total work done.

Key concepts

Spring Force Constant

In Hooke's Law, the spring force constant, often denoted as \( k \), describes how stiff or strong a spring is. It is a measure of the force required to compress or extend the spring by a unit distance. Given in units like Newtons per centimeter (N/cm), a higher spring constant signifies a stiffer spring that is harder to stretch or compress.
In practical scenarios, suppose a spring has a spring constant of 15 N/cm. For each centimeter you stretch it, you would need 15 Newtons of force. To calculate the work required in exercises or real-life situations, always ensure that your measurements are in the same units as your spring constant to avoid mistakes. Converting millimeters to centimeters, as in our original exercise, helps maintain consistency.

Work Done on a Spring

Work done on a spring is calculated using the formula:\[ W = \frac{1}{2} k X^2 \]where \( W \) is the work done, \( k \) is the spring constant, and \( X \) is the displacement from the spring's equilibrium position. The formula shows that work done is proportional to the square of the displacement, meaning the more you stretch or compress it, the more force is required.
For example, when calculating the work done to stretch a spring 7.60 mm, convert the displacement to 0.76 cm to match the spring constant's unit of N/cm. Insert these values into the formula to find the work. If you extend the spring further, compute the incremental work by calculating the work for the new displacement and subtracting the previous work done. This will give the additional energy required for further extensions.

Energy Stored in a Spring

When you do work on a spring, energy is stored in it as potential energy. This energy is described by the formula:\[ PE = \frac{1}{2} k X^2 \]The potential energy (PE) is equal to the work done for the displacement, showing a direct conversion from the work put in to energy stored. This energy is elastic potential energy because it's stored in the spring as it is compressed or stretched.
The energy can be calculated the same way as work, using the same spring constant and displacement. Understanding this concept is crucial, as it helps predict how much energy can be recuperated when the spring returns to its original state. Imagine compressing a spring deeply; it can release this stored energy, doing work on another object or reverting the spring to its equilibrium.