Question

If \((1 / 4)\) of a spring having length \(\ell\) is cutoff, then what will be the spring constant of remaining part? (A) \(\mathrm{k}\) (B) \(4 \mathrm{k}\) (C) \((4 \mathrm{k} / 3)\) (D) \((3 \mathrm{k} / 4)\)

Short answer

The short answer based on the step-by-step solution is: The spring constant of the remaining part of the spring will be \(\frac{16}{9}k\). However, none of the given options matches the correct answer. If the options were meant to have 16/9 rather than 4/3, then the correct option would be (C) \((16\mathrm{k} / 9)\).

Step-by-step solution

Find the new length of the spring

After cutting off one-quarter of the spring, the remaining length will be \( 3 \ell / 4\).

Write the equation for the relationship between the original and new spring constants

We know that the spring constant is inversely proportional to the length of the spring, so we can write the equation for the new spring constant k' as: \[ k' = k \cdot (\frac{4}{3})\cdot(\frac{\ell}{3\ell/4}) \]

Simplify the equation

Now we can simplify the equation as: \[k' = k \cdot (\frac{4}{3})\cdot(\frac{4}{3})\]

Solve for k'

Finally, we have the equation for the new spring constant k': \[k' = \frac{16}{9}k \] Comparing this expression to the four given options, we can see that none of them matches the correct answer. However, if the options were meant to have 16/9 rather than 4/3, then the correct option would be: (C) \((16\mathrm{k} / 9)\)

Key concepts

Spring Length

A spring's length refers to its full extent from one end to the other. When a spring is manipulated, often by cutting or stretching, its physical properties, such as spring constant, change.
For example:

• If a section of the spring is removed, the spring becomes shorter.
• This affects how the spring will behave under force.

A spring cut in length signifies the spring's ability to resist a force is redistributed. Typically, when a spring's length is reduced while maintaining its material properties, the spring constant (which measures the spring's stiffness) increases.
Hence, a shorter spring will normally exert more force for the same amount of displacement, showing an enhanced stiffness, which directly influences the point discussed in the Hooke's Law and proportionality concepts.

Hooke's Law

Hooke's Law is a principle of physics that dictates how a spring works when stretched or compressed. It states that the force needed to extend or compress a spring by a certain distance is proportional to that distance.
The mathematical representation of Hooke's Law is:\[ F = k imes x \]Where:

• \( F \): Force applied to the spring.
• \( k \): Spring constant.
• \( x \): Displacement of the spring from its rest position.

This equation shows that the larger the spring constant, \( k \), the stiffer the spring. That means it requires more force to achieve the same displacement. Applying Hooke's Law helps us to understand how the spring behaves under various conditions, such as when a section of the spring is cut off and the spring constant changes as a result.

Proportionality in Springs

Proportionality in springs refers to the relationship between variables such as force, displacement, and the spring's constant. Here:

• The spring constant is inversely proportional to the length of the spring. A shorter spring means a higher spring constant.
• Force is directly proportional to both the spring constant and the distance the spring is compressed or stretched, based on Hooke's Law.

When you cut a piece of the spring, you're not only changing its length but also its behaviors in these proportional relationships.
For instance, since the spring constant \( k \) is inversely related to length, reducing the length by cutting a portion of the spring leads to a larger spring constant, leading to a stiffer spring.
Understanding proportionality is crucial to solving problems where the length or other characteristics of springs are altered, allowing us to predict how the spring will respond under modified conditions.