You have two small, 2.0 g balls that have been given equal but opposite charges, but you don't know the magnitude of the charge. To find out, you place the balls distance apart on a slippery horizontal surface, release them, and use a motion detector to measure the initial acceleration of one of the balls toward the other. After repeating this for several different separation distances, your data are shown below. Use an appropriate graph of the data to determine the magnitude of the charge.

Verified step by step guidance

1. Step 1: Understand the problem. The balls are charged with equal but opposite charges, and their interaction is governed by Coulomb's law. The force between them causes acceleration, which is measured at different separation distances. The goal is to determine the magnitude of the charge using the data provided.
2. Step 2: Recall Coulomb's law, which states that the force between two charges is given by: $F=\frac{k{q}^2}{r^2}$, where $k$ is Coulomb's constant, $q$ is the magnitude of the charge, and $r$ is the separation distance.
3. Step 3: Relate the force to acceleration using Newton's second law: $F=ma$, where $m$ is the mass of the ball and $a$ is the acceleration. Combine this with Coulomb's law to express acceleration as: $a=\frac{k{q}^2}{m{r}^2}$.
4. Step 4: Plot the data. Create a graph with acceleration ($a$) on the y-axis and $\frac{1}{r^2}$ on the x-axis. The relationship between acceleration and $\frac{1}{r^2}$ should be linear, with the slope of the line equal to $\frac{k{q}^2}{m}$.
5. Step 5: Use the slope of the graph to calculate the magnitude of the charge. Rearrange the slope equation to solve for $q$: $q=\sqrt{\frac{\mathrm{slope}m}{k}}$. Substitute the known values for the slope, mass of the ball, and Coulomb's constant to find the charge.

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 relationship is crucial for understanding how the charges on the balls interact and how this interaction affects their acceleration.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle allows us to relate the measured acceleration of the charged balls to the electrostatic force acting on them, enabling the calculation of the charge magnitude when combined with Coulomb's Law.

Graphing and Data Analysis

Graphing the relationship between the distance between the balls and their acceleration is essential for visualizing the data. By plotting acceleration against distance, one can identify trends and derive a mathematical relationship that can be used to extract the charge magnitude. This process often involves fitting a curve to the data points to determine the constants involved in the equations derived from Coulomb's Law.