FIGURE EX5.8 shows an acceleration-versus-force graph for three objects pulled by rubber bands. The mass of object B is 0.20 kg. What are the masses of objects A and C? Explain your reasoning.

Verified step by step guidance

1. Step 1: Analyze the graph provided. The graph shows acceleration (y-axis) versus force (x-axis) for three objects (A, B, and C). The slope of each line represents the relationship between force and acceleration, which is inversely proportional to the mass of the object according to Newton's second law: $F=ma$.
2. Step 2: Use the given information for object B. The mass of object B is 0.20 kg. From the graph, the slope of the line for object B can be calculated as $a/F$. The slope is constant for each object and represents $1/m$. For object B, the slope is $5a/5$, which simplifies to $a$. This corresponds to $1/0.20$.
3. Step 3: Compare the slopes of the lines for objects A and C relative to object B. The slope for object A is steeper than that of object B, indicating a smaller mass (since $1/m$ is larger for smaller masses). The slope for object C is less steep than that of object B, indicating a larger mass.
4. Step 4: Calculate the relative masses. The slope for object A is approximately twice that of object B, meaning the mass of object A is half the mass of object B. Similarly, the slope for object C is approximately half that of object B, meaning the mass of object C is twice the mass of object B.
5. Step 5: Conclude the reasoning. Using the relationships derived from the graph and the given mass of object B (0.20 kg), the mass of object A is $0.10$ kg, and the mass of object C is $0.40$ kg.

Key Concepts

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 relationship is expressed by the formula F = ma, where F is the force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing the acceleration-versus-force graph, as it allows us to relate the forces applied to each object with their respective accelerations.

Slope of a Graph

In a force versus acceleration graph, the slope represents the mass of the object being analyzed. The steeper the slope, the smaller the mass, as a greater force results in a higher acceleration. By determining the slope of the lines for objects A and C in relation to object B, whose mass is known, we can calculate their respective masses based on the proportionality established by Newton's Second Law.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In the context of the graph, the acceleration of each object is proportional to the force applied, allowing us to compare the slopes of the lines for objects A, B, and C. By using the known mass of object B and its corresponding slope, we can derive the masses of objects A and C by establishing their ratios relative to B.