A 3.00-cm-long spring has a small plastic bead glued to each end. Charging each bead to −25 nC expands the spring by 0.50 cm. What is the value of the spring constant?

Verified step by step guidance

1. Step 1: Identify the forces acting on the spring. The spring is stretched due to the electrostatic repulsion between the two charged beads. The force due to the charges is given by Coulomb's law: $F=\frac{\left(k\right)\left(q_1{q}_2\right)}{r^2}$, where $k$ is Coulomb's constant, $q_1$ and $q_2$ are the charges, and $r$ is the distance between the charges.
2. Step 2: Write the expression for the spring force using Hooke's law: $F=k{x}_{\mathrm{spring}}$, where $k$ is the spring constant and $x_{\mathrm{spring}}$ is the extension of the spring (0.50 cm or 0.005 m).
3. Step 3: Set the magnitudes of the forces equal to each other because the system is in equilibrium: $\frac{\left(k\right)\left(q_1{q}_2\right)}{r^2}=k{x}_{\mathrm{spring}}$. Substitute the known values: $q_1=q_2=-25 \mathrm{nC}$ (convert to Coulombs: $-25\times {10}^{-}$ C), $r=3.50 \mathrm{cm}$ (convert to meters: $0.035$ m), and $x_{\mathrm{spring}}=0.005$ m.
4. Step 4: Rearrange the equation to solve for the spring constant $k$: $k=\frac{\left(k\right)\left(q_1{q}_2\right)}{r^2{x}_{\mathrm{spring}}}$. Substitute the numerical values for Coulomb's constant $k=8.99\times {10}^9$ N·m²/C², the charges, the distance, and the spring extension.
5. Step 5: Perform the calculation to find the spring constant $k$. Ensure all units are consistent (Coulombs, meters, and Newtons) and simplify the expression to determine the value of the spring constant.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its extension or compression from its equilibrium position, mathematically expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. This principle is fundamental for understanding how springs behave under applied forces.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects, stating that the force is proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is essential for calculating the force acting on the beads due to their charges, which in turn affects the spring's extension.

Spring Constant

The spring constant, denoted as k, is a measure of a spring's stiffness, indicating how much force is needed to stretch or compress the spring by a unit length. A higher spring constant means a stiffer spring that requires more force for the same displacement. It is a crucial parameter in both Hooke's Law and the analysis of spring systems.