Problem 19 - College Physics

Question

If it takes \(400 \mathrm{~J}\) of work to stretch a spring \(8.00 \mathrm{~cm}\) what is the spring constant?

Step-by-Step Solution

Answer

The spring constant is approximately 125000 N/m.

Step 1: Understanding Hooke's Law

Hooke's Law describes the behavior of springs, stating that the force needed to stretch or compress a spring is proportional to the distance it is stretched or compressed. This is expressed with 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 of the spring.

Step 2: Plugging in Given Values

We're given that the work \(W = 400 \text{ J}\), and the displacement \(x = 8.00 \text{ cm} = 0.08 \text{ m}\) (since we always convert to meters in SI units). The formula to find \(k\) becomes:\[ 400 = \frac{1}{2} k (0.08)^2 \]

Step 3: Solving for the Spring Constant

Now we will rearrange the formula to solve for \(k\). First, simplify the equation:\[ 400 = \frac{1}{2} k (0.0064) \]Multiply both sides by 2 to clear the fraction:\[ 800 = k (0.0064) \]Finally, divide both sides by 0.0064 to solve for \(k\):\[ k = \frac{800}{0.0064} \approx 125000 \, \text{N/m} \]

Step 4: Conclusion

After calculating, we find that the spring constant \(k\) is approximately \(125000 \text{ N/m}\). This value reflects how stiff or resistant to stretching the spring is.

Key Concepts

Spring Constant

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is needed to stretch or compress the spring by a certain amount. Think of it as the difficulty level in changing the length of the spring. A higher spring constant means the spring is stiffer, making it harder to stretch. In our example, the spring constant was calculated to be \( 125000 \, \text{N/m} \). This value is obtained using Hooke's Law, which mathematically expresses this concept as \( W = \frac{1}{2} k x^2 \). From this formula, \( k \) can be isolated and calculated if the work \( W \) and displacement \( x \) are known.

When dealing with springs, always remember:

• Higher \( k \): Stiffer spring
• Lower \( k \): Softer spring
• Always express displacement in meters for accuracy

This concept is crucial in understanding how different materials and structures respond to forces.

Work and Energy

Work and energy are key concepts when analyzing springs. Work, in physics, is the amount of energy transferred by a force through a distance. In our exercise, it took \( 400 \, \text{J} \) (joules) of work to stretch the spring.

This means energy from an external force was applied to change the spring's length. The formula \( W = \frac{1}{2} k x^2 \) links work done on the spring to its displacement and spring constant. This equation shows that:

• More work equals more stretch
• Stiffer springs require more work for the same stretch
• Energy is stored in the spring, a concept known as potential energy

Understanding work and energy flow is essential in systems involving springs, such as mechanical watches or car suspensions.

Displacement

Displacement in the context of Hooke’s Law is the change in length of a spring when a force is applied. In our scenario, the spring was stretched by \( 8.00 \, \text{cm} \), which converts to \( 0.08 \, \text{m} \). Always remember to convert any measurements to meters in physics for consistency with SI units.

Displacement is a pivotal factor in Hooke's Law, as it directly affects how much work is done on the spring. The relationship can be observed in:

• Small displacement = less work
• Large displacement = more work
• The displacement squared factor indicates non-linear impact on energy needed

This squared factor in the equation \( W = \frac{1}{2} k x^2 \) means that even small changes in displacement can lead to large differences in work done, illustrating the significant impact of displacement in spring physics.