Question

A certain force gives object \(m_{1}\) an acceleration of \(12.0 \mathrm{~m} / \mathrm{s}^{2}\) The same force gives object \(m_{2}\) an acceleration of \(3.30 \mathrm{~m} / \mathrm{s}^{2}\) What acceleration would the force give to an object whose mass is \((a)\) the difference between \(m_{1}\) and \(m_{2}\) and \((b)\) the sum of \(m_{1}\) and \(m_{2}\) ?

Short answer

The acceleration given to an object whose mass is (a) the difference between \(m_{1}\) and \(m_{2}\) would be \(a_{1} / (1 - (12/3.3))\), and (b) the sum of \(m_{1}\) and \(m_{2}\) would be \(a_{1} / (1 + (12/3.3))\).

Step-by-step solution

Determine the Ratio of the Masses

First, use Newton's Second Law to equate the two forces involved: \(F = m_1a_1 = m_2a_2\). This gives us the ratio \(m_{2}/m_{1} = a_{1}/a_{2} = 12/3.3\).

Calculate Mass Difference

For (a) m_{1 - 2} = m_{1} - m_{2}, we'll use the ratio to determine the result. This gives us that m_{1 - 2} is equivalent to one object of \(m_{1}\) minus (12/3.3) objects of \(m_{1}\), which gives m_{1 - 2} = \(m_{1}(1 - (12/3.3))\). The acceleration a_{1 - 2} of the new object would then be \(F / m_{1 - 2}\). We can simplify by noting the force on each object, \(F = m_{1}a_{1} = m_{2}a_{2}\), so the above equation gives us: \(a_{1 - 2} = a_{1} / (1 - (12/3.3))\).

Calculate Mass Sum

For (b) m_{1 + 2} = m_{1} + m_{2}, we do a similar process. This is equivalent to one object of \(m_{1}\) plus (12/3.3) objects of \(m_{1}\), which gives m_{1 + 2} = \(m_{1}(1 + (12/3.3))\). The acceleration a_{1 + 2} of the new object and force simplifies similarly, we find: \(a_{1 + 2} = a_{1} / (1 + (12/3.3))\).

Key concepts

Classical Mechanics

Classical mechanics is a fundamental part of physics that deals with the motion of objects and the forces that affect that motion. It is grounded in Newton's three laws of motion, which describe the relationship between a body and the forces acting upon it, and the body's response in terms of movement. Understanding classical mechanics is essential for solving a wide variety of physics problems, ranging from the simplest cases of linear motion to more complex scenarios involving multiple forces.

For students tackling problems in classical mechanics, it's important to visualize the situation, breaking down complex motions into simpler, component parts and applying Newton's laws to each part. Problems often involve calculating forces, masses, and accelerations, creating mathematical models that predict how an object will move under certain conditions.

Force and Acceleration

The cornerstone of classical mechanics is Newton's Second Law, which states that the force applied to an object is equal to the mass of the object multiplied by its acceleration (\( F = ma \)). This highlights a direct proportionality between force and acceleration, indicating that for a constant force, an object's acceleration is inversely proportional to its mass.

In practice, when the same force is applied to two different objects, the lighter object will experience a greater acceleration. Students should remember that when solving problems involving force and acceleration, it is often helpful to isolate these variables. This simplification allows for the direct comparison of the effect of the force on different masses, as seen in the textbook exercise.

Mass Ratio Calculation

To understand mass ratio calculation in the context of physics problems, consider two objects subject to the same force resulting in different accelerations. By comparing these accelerations, we can determine the ratio of their masses without knowing their exact mass values. This technique is a powerful tool and is particularly useful in situations where objects interact with each other or are compared under similar conditions.

The formula derived from Newton's Second Law, wherein the mass of one object over the mass of another equals the acceleration of the second object over the acceleration of the first (\( m_2/m_1 = a_1/a_2 \)), allows us to perform these calculations. Understanding how to use mass ratios can simplify complex problems, making it easier to solve for unknowns through proportional reasoning.

Physics Problem Solving

Physics problem solving is an intellectual exercise that requires a methodical approach to understand and apply physical principles to find a solution. It involves several steps: identifying the relevant principles, developing the appropriate mathematical model, and performing calculations to solve for unknown quantities.

In the given textbook exercise, the solution starts by identifying Newton's Second Law as relevant. Next, it models the problem through the lens of force and mass interactions. Calculating the mass ratio is critical to find the acceleration for different mass combinations. Such problems can embody various physics concepts and often necessitate assumptions and approximations. To enhance their problem-solving skills, students should practice approaching problems from different angles, checking units for consistency, and validating the reasonableness of their answers.