Problem 8 - Essential Calculus

Question

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring I foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet and the springs are at their natural length when the door is open. Find the work done by the pair of springs.

Step-by-Step Solution

Answer

The work done by the pair of springs when the door moves 8 feet is 240 foot-pounds.

Step 1: Determine the Spring Constant

From the problem, it is understood that a force of 15 pounds stretches each spring by 1 foot. Therefore, the spring constant \(k\) is 15 lbs/ft.

Step 2: Calculate the Distance Each Spring Stretches

The door moves a total of 8 feet and because of the pulley system, the springs stretch only half the distance. Therefore, the distance each spring stretches (\(x\)) is 8 / 2 = 4 feet.

Step 3: Calculate the Work Done by One Spring

Work done to stretch or compress a spring is given by \(W = \frac{1}{2}kx^2\). Substituting the values for \(k\) (15 lbs/ft) and \(x\) (4 ft), we get \(W = \frac{1}{2} * 15 * (4^2) = 120\) foot-pounds.

Step 4: Compute the Total Work Done by Both Springs

Since there are two springs and they both do the same amount of work, the total work done is twice the work done by one spring. Therefore, the total work is \(2 * W = 2 * 120 = 240\) foot-pounds.

Key Concepts

Spring Constant

The spring constant, often represented by the symbol \( k \), is a measure of the stiffness or rigidity of a spring. In Hooke's Law, it quantifies the relationship between the force applied to a spring and the amount that the spring stretches or compresses. The formula is given by:

• \( F = kx \)

where \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement of the spring from its natural length.

In our example, a force of 15 pounds stretches each spring by 1 foot. This means that for every foot the spring is stretched, a force of 15 lbs is required. Thus, the spring constant \( k \) is 15 lbs/ft. The spring constant is crucial as it helps to predict how a spring will behave under different forces, providing insight into both everyday applications like garage doors and complex engineering problems.

Work Done by Spring

Work done by a spring involves calculating the energy required to stretch or compress the spring from its natural position. This can be determined using the formula:

• \( W = \frac{1}{2}kx^2 \)

where \( W \) is the work done, \( k \) is the spring constant, and \( x \) is the amount the spring is stretched or compressed.

In our garage door example, each spring stretches 4 feet when the door moves 8 feet due to the pulley system. Given the spring constant of 15 lbs/ft, we compute the work done by one spring as:

• \( W = \frac{1}{2} \times 15 \times (4^2) = 120 \) foot-pounds

This value represents the energy stored in the spring when stretched. For two springs, as in the overhead garage door, the total work done is simply twice the amount of one spring's work, resulting in 240 foot-pounds.

Force and Distance in Springs

The relationship between force and distance in springs is summed up by Hooke's Law. As explored earlier, this law states that the force required to either compress or stretch a spring is directly proportional to the distance the spring is moved from its equilibrium position.

• Force Formula: \( F = kx \)

The force is determined by multiplying the spring constant \( k \) by the distance \( x \).

In scenarios involving garages, the practical application is straightforward. For every movement of the garage door, the springs experience only half the distance covered by the door due to the pulley mechanism. Hence, when the garage door moves 8 feet, each spring stretches just 4 feet. This direct relationship allows us to predict how a given force will stretch the springs and is essential for accurately calculating the work done by or stored in the springs.