Problem 54 - College Physics

Question

A horizontal force of \(40 \mathrm{~N}\) acting on a block on a frictionless, level surface produces an acceleration of \(2.5 \mathrm{~m} / \mathrm{s}^{2} .\) A second block, with a mass of \(4.0 \mathrm{~kg}\), is dropped onto the first. What is the magnitude of the acceleration of the combination of blocks if the same force continues to act? (Assume that the second block does not slide on the first block.)

Step-by-Step Solution

Answer

The acceleration of the combined blocks is 2 m/s².

Step 1: Determine the Mass of the First Block

Use Newton's second law, \( F = ma \), where \( F = 40 \text{ N} \) and \( a = 2.5 \text{ m/s}^2 \), to find the mass \( m \) of the first block. \[ m = \frac{F}{a} = \frac{40 \, \text{N}}{2.5 \, \text{m/s}^2} = 16 \, \text{kg} \] Hence, the mass of the first block is 16 kg.

Step 2: Calculate the Total Mass of Both Blocks

Add the mass of the first block to the mass of the second block, which is 4 kg. \[ \text{Total mass} = 16 \, \text{kg} + 4 \, \text{kg} = 20 \, \text{kg} \] This is the combined mass of both blocks.

Step 3: Use Newton's Second Law on Combined System

Apply Newton's second law to the combination of blocks. The force \( F \) remains the same at 40 N, but now the total mass is \( 20 \, \text{kg} \). The formula is \( F = ma \), so rearrange it to solve for \( a \): \[ a = \frac{F}{m} = \frac{40 \, \text{N}}{20 \, \text{kg}} = 2 \, \text{m/s}^2 \] Thus, the acceleration of the combined system is 2 m/s².

Key Concepts

Force and Acceleration

Newton's Second Law of Motion is a fundamental principle in physics. It tells us how the motion of an object changes when a force is applied. The formula is expressed as \( F = ma \). Here, \( F \) is the force applied, \( m \) is the mass of the object, and \( a \) is the acceleration produced by the force.
When the force remains constant and the mass changes, the acceleration changes as well. This is what happens when you add more mass onto an existing object while maintaining the same force.

• If the mass increases, and the force stays the same, the acceleration decreases.
• If the mass decreases, and the force stays the same, the acceleration increases.

In the exercise, we initially had a block experiencing a force of 40 N, which produced an acceleration of 2.5 m/s². By using the formula \( a = \frac{F}{m} \), we found the acceleration when the mass was increased due to adding a second block.

Mass and Weight

Understanding the concepts of mass and weight is crucial in grasping the essence of forces in physics. Mass refers to the amount of matter in an object and is measured in kilograms (kg). It's a scalar quantity, meaning it only has magnitude and no direction.
In contrast, weight is the force exerted by gravity on an object, which depends on both mass and the gravitational acceleration, typically \( 9.8 \text{ m/s}^2 \) on Earth. If someone asks for your weight, they are asking for the force gravity exerts on your mass.

• Mass remains the same regardless of where you are in the universe.
• Weight can change depending on gravitational pull. For example, on the moon, you'd weigh less than on Earth.

In the problem, the mass of the first block was calculated as 16 kg, and the second block added 4 kg to the system. These measurements are crucial to determining the total mass in motion when applying Newton's Second Law of Motion in the problem.

Physics Problem Solving

Breaking down physics problems into manageable steps is vital for success in problem-solving. A systematic approach helps uncover complex underlying physics concepts.
Here's how you can efficiently tackle physics problems:

• First, identify known quantities: force, mass, and acceleration.
• Apply relevant physics laws: In this case, Newton's Second Law \( F = ma \).
• Solve step by step: Find any unknown values using calculations, like finding mass and then recalculating acceleration.
• Always check your units: Ensure calculations are consistent with units of force (N), mass (kg), and acceleration (m/s²).

In the problem provided, knowing Newton's Second Law helped break down the scenario into calculable steps. By first finding the mass of the first block, adding the second, and calculating the combined acceleration, you arrive at the solution efficiently.