A massless spring is attached to a support at one end and has a 2.0 μC charge glued to the other end. A −4.0 μC charge is slowly brought near. The spring has stretched 1.2 cm when the charges are 2.6 cm apart. What is the spring constant of the spring?

Verified step by step guidance

1. Identify the forces acting on the system. The spring is stretched due to the electrostatic force between the two charges. The electrostatic force is given by Coulomb's law: $F=\frac{k{e}_{}\cdot q_1\cdot q_2}{r^2}$, where $q_1$ and $q_2$ are the charges, $r$ is the distance between them, and $k_e$ is Coulomb's constant ($k_e=8.99\times 109N·m²/C²$).
2. Relate the electrostatic force to the spring force. The spring force is given by Hooke's law: $F=k\cdot x$, where $k$ is the spring constant and $x$ is the displacement of the spring. At equilibrium, the spring force equals the electrostatic force: $k_e\cdot \frac{q_1}{q_2}=k\cdot x$.
3. Substitute the known values into Coulomb's law. Use $q_1=2.0\times 10-6C$, $q_2=-4.0\times 10-6C$, $r=2.6\times 10-2m$, and $k_e=8.99\times 109N·m²/C²$ to calculate the electrostatic force: $F=\frac{k_e\cdot q_1\cdot q_2}{r^2}$.
4. Set the electrostatic force equal to the spring force. Use the displacement of the spring, $x=1.2\times 10-2m$, and solve for the spring constant $k$: $k=\frac{F}{x}$.
5. Simplify the expression to find the spring constant. Substitute the value of the electrostatic force from step 3 and the displacement of the spring into the equation from step 4 to calculate $k$.

Key Concepts

Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force is directly 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 spring due to the charges.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to its extension or compression from its equilibrium position. Mathematically, it is expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement. This principle is crucial for determining the spring constant in the context of the problem.

Equilibrium of Forces

In this scenario, the system reaches equilibrium when the electrostatic force exerted by the charges equals the restoring force of the spring. Understanding this balance is key to solving for the spring constant, as it allows us to set the forces equal to each other and solve for the unknown variable.