Question

The force \(F\) a spring exerts on you is directly proportional to the distance \(x\) you stretch it beyond its resting length. Suppose that when you stretch a spring \(8.00 \mathrm{~cm},\) it exerts a 200. N force on you. How much force will it exert on you if you stretch it \(40.0 \mathrm{~cm} ?\)

Short answer

Answer: The force exerted on the user if the spring is stretched to 40 cm is 1000 N.

Step-by-step solution

Setup the equation

The given equation is \(F = kx\). Given that \(F=200\) N and \(x=0.08\) m, we can setup the equation as: \(200 = k \times 0.08\).

Solve for the proportionality constant (k)

Rearrange the equation to solve for k: \(k = \frac{200}{0.08}\). After calculating, we get \(k = 2500\) N/m.

Use the proportionality constant to find the force exerted by the spring when it is stretched to 40 cm

In this step, we will use the constant k to find the force exerted by the spring when it is stretched to 40 cm. We have \(F = kx\). Convert 40 cm to meters (1 cm = 0.01 m): \(x = 0.4\) m. Then plug in the values: \(F = 2500 \times 0.4\).

Calculate the force

After calculating, we get the force as: \(F = 2500 \times 0.4 = 1000\) N. Final Answer: The force exerted on the user if the spring is stretched to 40 cm is 1000 N.

Key concepts

Understanding the Force and Displacement Relationship

When dealing with springs and their forces, it's crucial to understand the link between force and displacement. This relationship follows Hooke's Law, stating that the force exerted by a spring is directly proportional to the displacement from its resting position. In simpler terms, the further you stretch the spring, the greater the force it exerts to return to its original position.
Imagine a spring in its natural, unaltered state—this is referred to as its resting length. If you pull on the spring, extending it by a certain distance, this distance is called displacement, denoted as \( x \). The spring then responds by exerting a force, noted as \( F \), to resist this pulling.

• The direction of the force is always opposite to the direction of displacement, meaning the spring pulls back when stretched and pushes back when compressed.
• This concept implies a linear relationship, suggesting that doubling the displacement doubles the force.

In our given problem, stretching the spring by 8 cm exerts a 200 N force. Thus, it's crucial to measure displacement accurately to determine the force according to Hooke’s Law.

Proportionality Constant and its Significance

In Hooke's Law, the concept of a proportionality constant, represented as \( k \), plays a vital role. This constant is known as the spring constant, and it helps determine the relationship between force and displacement. The equation \( F = kx \) clarifies that \( k \) is a unique value that converts displacement into the exerted force.
The spring constant \( k \) measures the stiffness of a spring. A higher value indicates a stiffer spring that requires more force to be displaced by a certain length.

• If two springs have different \( k \) values, the one with the higher \( k \) will exert more force for the same displacement.
• The spring constant is determined by the properties of the spring material and its construction.

In our example, by rearranging the formula to solve for \( k \) using given values, we found \( k = 2500 \text{ N/m} \). This means for every meter the spring is stretched, it exerts 2500 N of force.

Spring Constant Calculation in Action

Calculating the spring constant \( k \) allows us to predict how much force the spring will exert for any given displacement, which illustrates the power of understanding Hooke's Law.
Once you've calculated \( k \), you can easily determine the force for any amount of stretch. The steps involve:

• Convert the displacement from centimeters to meters to maintain consistent units.
• Apply the equation \( F = kx \) where both \( k \) and \( x \) are in their respective units.
• Multiply to find the resulting force.

In the problem we solved, we initially calculated \( k \) as mentioned earlier. Using \( k = 2500 \text{ N/m} \) and stretching the spring by 0.4 m (converted from 40 cm), we applied the formula to find \( F = 2500 \times 0.4 = 1000 \text{ N} \). Thus, if the spring is stretched to 40 cm, it exerts 1000 N of force. This calculation demonstrates the practical utility of finding the spring constant, enabling us to accurately predict the force exerted by the spring for various displacements.