Problem 13 - College Physics

Question

A force acts on a \(1.5-\mathrm{kg},\) mass, giving it an acceleration of \(3.0 \mathrm{~m} / \mathrm{s}^{2} .\) (a) If the same force acts on a 2.5 -kg mass, what acceleration would be produced? (b) What is the magnitude of the force?

Step-by-Step Solution

Answer

(a) The acceleration is 1.8 m/s². (b) The force magnitude is 4.5 N.

Step 1: Understanding the relationship between force, mass, and acceleration

The force acting on an object can be calculated using Newton's second law of motion, which states that force (\( F \)) is the product of mass (\( m \)) and acceleration (\( a \)). This is represented by the equation \( F = m \times a \).

Step 2: Calculate the force using the initial conditions

Given the mass \( m = 1.5 \, \text{kg} \) and acceleration \( a = 3.0 \, \text{m/s}^2 \), we can calculate the force:\[ F = 1.5 \, \text{kg} \times 3.0 \, \text{m/s}^2 \]\[ F = 4.5 \, \text{N} \]

Step 3: Use the calculated force for the new mass to find acceleration

Now, using the force \( F = 4.5 \, \text{N} \) calculated in Step 2, find the acceleration for the second mass, \( m = 2.5 \, \text{kg} \), using the formula:\[ a = \frac{F}{m} \]\[ a = \frac{4.5 \, \text{N}}{2.5 \, \text{kg}} \]\[ a = 1.8 \, \text{m/s}^2 \]

Step 4: Finding the magnitude of the force

The magnitude of the force was already determined in Step 2. It is \( 4.5 \, \text{N} \).

Key Concepts

Force Calculation

Newton's Second Law of Motion offers a straightforward approach to understanding how forces influence motion. At its core, the law reveals that force is the result of multiplying mass and acceleration.
In equation form, it is represented as:

• \( F = m \times a \)

Given information like mass and acceleration, it's simple to calculate the force exerted. Suppose you have a mass of \(1.5 \, \text{kg}\) accelerating at \(3.0 \, \text{m/s}^2\). Plug these values into the formula:

• \( F = 1.5 \, \text{kg} \times 3.0 \, \text{m/s}^2 \)
• \( F = 4.5 \, \text{N} \)

This equation and calculation forms the basis of solving many force-related problems in physics.

Mass and Acceleration Relationship

The interplay between mass and acceleration is key in understanding motion dynamics. One should realize that with a constant force, any increase in mass results in a proportional decrease in acceleration. This inverse relationship can be illustrated when a force acts differently on different masses.
In the given exercise, a force of \(4.5 \, \text{N}\) is initially calculated for a \(1.5 \, \text{kg}\) mass. If this same force is applied to a \(2.5 \, \text{kg}\) mass, find the new acceleration by using:

• \( a = \frac{F}{m} \)
• \( a = \frac{4.5 \, \text{N}}{2.5 \, \text{kg}} \)
• \( a = 1.8 \, \text{m/s}^2 \)

This shows how an increased mass results in a reduced acceleration when the same force is applied.

Physics Problem-Solving

When addressing physics problems, breaking them into manageable parts can simplify complex scenarios. Here’s a structured approach to attack such problems effectively:

• **Identify Known Variables:** Gather given data such as mass, acceleration, or force.
• **Apply Newton's Second Law:** Use \( F = m \times a \) to find unknown quantities.
• **Use Consistent Units:** Make sure all measurements are in compatible units (e.g., kilograms for mass, meters per second squared for acceleration).
• **Solve Step-by-Step:** Start by finding one variable before moving on to others.

In this particular exercise, having already calculated the force, we utilized that calculated force to find how acceleration varies with different mass substitutions. Such a step-by-step approach aids in clear understanding and minimizes errors.