Question

Hooke's Law Hooke's Law also applies to compressing springs; that is, it requires a force of \(k x\) to compress a spring a distance \(x\) from its natural length. Suppose a \(10,000-\) -lb force compressed a spring from its natural length of 12 inches to a length of 11 inches. How much work was done in compressing the spring (a) the first half-inch? (b) the second half-inch? You may use a graphing calculator to solve the following problems.

Short answer

The work done in compressing the spring for the first half-inch is 1250 lb-in and for the second half-inch is 3750 lb-in.

Step-by-step solution

Calculate the Constant 'k'

According to Hooke's Law, Force = k * x. We can calculate 'k' using this formula. Given that the Force is 10000 lb and it compressed the spring from its natural length of 12 inches to a length of 11 inches. Therefore, 'x' equals to 1 inch. We can rewrite the formula to find 'k' as k = Force / x. Substituting the values, k = 10000 / 1 = 10000 lb/inch.

Find Work Done for the First Half-Inch

For the first half inch compression, we'll use the formula Work = 1/2 * k * x². Here x=0.5. So, Work = 1/2 * 10000 * (0.5)² = 1250 lb-in.

Find Work Done for the Second Half-Inch

For the second half inch, we still use the formula Work = 1/2 * k * x². But this time 'x' is the total distance compressed (1 inch), not just the additional distance. Thus, we subtract the work done in the first half inch from total work done in compressing the spring by 1 inch. Work for one inch compression = 1/2 * 10000 * (1)² = 5000 lb-in. Work for the second half-inch = Work for one inch - Work for the first half inch = 5000 lb-in - 1250 lb-in = 3750 lb-in.

Key concepts

Spring Compression

When you press or squeeze a spring, reducing its length, you're engaged in spring compression. Hooke's Law provides a way to quantify this action, stating that the force needed to compress or extend a spring is directly proportional to the distance the spring is stretched or compressed from its resting position. This distance is often noted as 'x'.

Imagine you have a spring that doesn't change shape; applying a force to this spring pushes it back with equal force. However, in the real world, springs compress, and when this happens, the energy is neither lost nor gained - it is stored in the spring as potential energy. The bigger the force you apply, the more the spring compresses, and the more energy it stores up to a limit before the material could be deformed.

In our exercise scenario, compressing the spring by one inch with a 10,000 lb force illustrates Hooke's Law in action. We refer to the initial, uncompressed length of the spring as its natural length. When you compress the spring, measuring the work done at different intervals can reveal interesting insights about energy storage and distribution within the spring.

Work Done

Whenever force is applied to an object causing it to move - or in the case of springs, compress - work is done. In physics, the work done on an object is the amount of energy transferred to or from an object via the application of force along a displacement. Significantly, work is not just about exertion but displacement and force in tandem.

The formula to calculate work done in compressing or stretching a spring is derived from Hooke's Law and can be expressed as \( W = \frac{1}{2} k x^2 \), where 'W' represents work, 'k' is the spring's force constant, and 'x' is the displacement of the spring from its natural length.

Using this formula, the step-by-step solution of our exercise indicates that more work is required to compress the spring during the second half-inch than the first. This is because as the spring compresses, it takes more and more energy to compress it further due to the increasing opposition of the spring's restoring force.

Force Constant k

The force constant k, integral to Hooke's Law, represents the stiffness of a spring. The higher the force constant, the stiffer the spring, and the more force it will take to compress or extend it by a certain amount. Essentially, 'k' provides a measure of the spring's resistance to compression or extension.

To determine the force constant 'k', you can use the formula derived from Hooke's Law: \( k = \frac{F}{x} \), where 'F' is the force applied to the spring, and 'x' is the displacement caused by the force. In the context of our exercise, the force constant 'k' is found by dividing the applied force (10000 lb) by the displacement from the natural length of the spring (1 inch), yielding a force constant of 10000 lb/inch.

This value is crucial for solving the subsequent parts of the problem, such as calculating the work done at different stages of spring compression. It helps us understand that the 'stiffer' the spring (reflected in a higher value of 'k'), the greater the amount of work needed for a given displacement.

Graphing Calculator

A graphing calculator is a powerful tool that can simplify complex mathematical problems, particularly those involving variables and equations. It allows users to visualize functions and their characteristics, aiding in the understanding of mathematical concepts.

In exercises involving Hooke's Law, a graphing calculator can prove invaluable. By inputting the formula for work done or force against displacement, students can create graphs demonstrating the relationship between force applied and displacement of a spring. These graphs typically show a linear relationship as predicted by Hooke's Law—at least up until the point of elastic limit, beyond which the spring may deform irreversibly.

Furthermore, a graphing calculator can help with the computation of work done at various points as the spring is compressed or extended. It can quickly calculate and graph the total work done without manually inputting and solving numerous equations, thus demonstrating the practical applications of Hooke's Law over different intervals of force and displacement.