Problem 20 - College Physics

Question

A force of \(50 \mathrm{~N}\) acts on a mass \(m_{1}\), giving it an acceleration of \(4.0 \mathrm{~m} / \mathrm{s}^{2}\). The same force acts on a mass \(m_{2}\) and produces an acceleration of \(12 \mathrm{~m} / \mathrm{s}^{2} .\) What acceleration will this force produce if the total system is \(m_{1}+m_{2}\) ?

Step-by-Step Solution

Answer

The combined acceleration is approximately \(3.00 \mathrm{~m/s^2}\).

Step 1: Understand the Given Data and Equations

We have two masses, \( m_1 \) and \( m_2 \), with a force \( F = 50 \mathrm{~N} \) acting on each separately. The acceleration produced on mass \( m_1 \) is \( a_1 = 4.0 \mathrm{~m/s^2} \) and on \( m_2 \) is \( a_2 = 12.0 \mathrm{~m/s^2} \). We need to determine the combined acceleration when both masses are subjected to the same force together. This requires using Newton's second law \( F = ma \).

Step 2: Calculate Individual Masses

Using \( F = ma \), solve for each mass: \( m_1 = \frac{F}{a_1} = \frac{50}{4.0} = 12.5 \mathrm{~kg} \) \( m_2 = \frac{F}{a_2} = \frac{50}{12.0} \approx 4.17 \mathrm{~kg} \).

Step 3: Calculate the Total Mass of the System

Add the masses of \( m_1 \) and \( m_2 \) to find the total mass: \( m_{total} = m_1 + m_2 = 12.5 + 4.17 = 16.67 \mathrm{~kg} \).

Step 4: Calculate the Acceleration of the Combined System

Use the formula \( a = \frac{F}{m_{total}} \) to calculate acceleration: \( a_{combined} = \frac{50 \mathrm{~N}}{16.67 \mathrm{~kg}} \approx 3.00 \mathrm{~m/s^2} \).

Key Concepts

Force

In physics, force is any interaction that, when unopposed, changes the motion of an object. A simple way to think about force is by considering it a push or a pull upon an object resulting from its interaction with another object. Understanding force is crucial because it helps us to predict how objects will move. According to Newton's Second Law of Motion, force is directly proportional to both the mass of the object and the acceleration it undergoes. Mathematically, this relationship is expressed as:\[ F = ma \]where:- \( F \) is the force applied,- \( m \) is the mass of the object, and- \( a \) is the acceleration.This equation shows that for a constant force, an increase in mass results in a decrease in acceleration, and vice versa. This principle underlies much of classical mechanics and is critical for problems involving multiple objects and their movements in daily physics applications.

Acceleration

Acceleration refers to the rate of change of velocity of an object. It is a vector quantity, which means it has both a magnitude and a direction. In straightforward terms, when an object speeds up, slows down, or changes direction, it is accelerating. The unit of acceleration is meters per second squared (\( ext{m/s}^2 \)), which describes how quickly velocity changes each second.In the context of Newton's Second Law, acceleration is directly influenced by both the amount of force applied and the mass of the object. If the same force acts on two objects of different masses, the acceleration will differ due to their masses. As seen in the problem scenario:- Mass \( m_1 \) experiences an acceleration \( a_1 = 4.0 ext{ m/s}^2 \).- Mass \( m_2 \) experiences an acceleration \( a_2 = 12.0 ext{ m/s}^2 \).From these observations, you can deduce that a smaller mass will accelerate more under the same force. This concept assists in calculating unknown variables when dealing with multiple masses and forces in physics.

Mass

Mass is a fundamental property of physical objects that determines their resistance to being accelerated or decelerated when a force is applied. In other words, it is a measure of an object's inertia. The greater the mass of an object, the less acceleration it will experience for a given force. Mass is typically measured in kilograms (\( ext{kg} \)).From the given exercise, individual masses were found using the formula derived from Newton's Second Law:\[ m = \frac{F}{a} \]For the initial calculations:- Mass \( m_1 = 12.5 \text{ kg} \) because it is subjected to the force of \( 50 ext{ N} \) resulting in an acceleration of \( 4.0 ext{ m/s}^2 \).- Mass \( m_2 = 4.17 \text{ kg} \) due to the same force causing an acceleration of \( 12.0 ext{ m/s}^2 \).Understanding mass in the context of forces allows you to predict how different objects react under the same or varying forces. It highlights the importance of calculating total mass when determining the system's collective behavior, such as the combined acceleration of \( m_1 + m_2 \).