Physics Problem: Masses On A String Explained

Hey there, physics enthusiasts! Today, we're diving into a classic physics problem that's super common: Two masses hanging on a string. We'll break it down step-by-step, making sure everyone understands, even if physics isn't your favorite thing. We are going to explore this problem by considering the forces acting on each mass and then solving for the acceleration of the system and the tension in the string. It is a fundamental concept in mechanics, often encountered in introductory physics courses. Understanding how to solve this problem is crucial for grasping more complex dynamics problems later on.

Let's start with the basics. Imagine a massless, ideal string (meaning it has no weight and doesn't stretch). We hang two masses from this string. One mass is $200 ext{ g}$, and the other is $600 ext{ g}$. The setup might seem simple, but it brings up some cool physics concepts. What's the acceleration of the masses? What is the tension in the string? These are the questions we'll answer.

First of all, let's look at the forces. Gravity is the big player here, pulling both masses downwards. The string is also acting, but the string's role is to transmit the force, not to provide an additional force. The tension in the string is constant throughout, which is important. The mass is the other factor to consider. So, understanding the free-body diagram of each mass is also important. The free-body diagram shows us all the forces acting on each object. For the lighter mass, the force due to gravity ($F_g$) will be less, and the tension in the string ($T$) will be greater. This means that if we are to solve this problem then we need to set up a coordinate system. Let's call the direction of the heavier mass's acceleration positive. The free body diagram, will help you, so I encourage you to use one in your own study. We'll use Newton's Second Law ($F = ma$) to figure out what's going on. This law connects forces, mass, and acceleration, and it's our key to unlocking the problem. The heavier mass will accelerate downwards, and the lighter mass will accelerate upwards. This movement is due to the difference in the forces of gravity acting on each mass. We'll use the principles of Newton's second law to solve for the acceleration of the system, and the tension of the string.

Setting Up the Problem and Defining Variables

Alright, let's get down to business, guys! To tackle this problem, we need to first set things up properly. It's like setting the stage before a play. We'll define some variables and establish our coordinate system. This is crucial for keeping track of everything and avoiding confusion. Here's what we'll do:

• Masses: Let's denote the smaller mass as $m_1 = 200 ext{ g} = 0.2 ext{ kg}$ and the larger mass as $m_2 = 600 ext{ g} = 0.6 ext{ kg}$. Remember, we always convert grams to kilograms because we're working in the standard SI units.
• Acceleration due to gravity: This is a constant, usually denoted as $g = 9.8 ext{ m/s}^2$. This is the acceleration that each mass would experience if it were falling freely.
• Tension in the string: We'll denote this as $T$. The tension is the force exerted by the string on each mass, pulling upwards on the lighter mass and upwards (opposing the gravitational pull) on the heavier mass.
• Acceleration of the system: We'll denote this as $a$. Since the masses are connected by the string, they'll both have the same magnitude of acceleration, but in opposite directions. The heavier mass will accelerate downwards, and the lighter mass will accelerate upwards.

Now, about the coordinate system. For convenience, let's define the downwards direction as positive for $m_2$ and the upwards direction as positive for $m_1$. This means that $m_2$ will have a positive acceleration, and $m_1$ will have a negative acceleration (since it's accelerating upwards). This choice is arbitrary, but it helps keep our equations organized. Having these variables and a coordinate system will help you keep track of all the forces in the system. The next step is to draw the free body diagram, and then the final step is to apply Newton's second law. This will allow you to solve for the acceleration of the system, and the tension in the string.

Drawing Free-Body Diagrams and Applying Newton's Second Law

Okay, now that we have our variables defined and our coordinate system set, it's time to get into the heart of the problem. We'll use free-body diagrams to visualize the forces acting on each mass. A free-body diagram is a simple sketch showing all the forces acting on an object. This is a crucial step in solving any dynamics problem. Let's draw the free-body diagrams for $m_1$ and $m_2$.

• Free-Body Diagram for $m_1$: This mass has two forces acting on it: the force of gravity, $F_{g1} = m_1g$, acting downwards, and the tension in the string, $T$, acting upwards.
• Free-Body Diagram for $m_2$: This mass also has two forces acting on it: the force of gravity, $F_{g2} = m_2g$, acting downwards, and the tension in the string, $T$, acting upwards.

Now, we'll apply Newton's Second Law ($F = ma$) to each mass. Remember, this law states that the net force acting on an object is equal to its mass times its acceleration. Let's write the equations for each mass:

• For $m_1$: Since we defined upwards as positive for $m_1$, the net force is $T - m_1g = m_1a$. Notice that the tension is positive because it acts in the positive direction (upwards).
• For $m_2$: Since we defined downwards as positive for $m_2$, the net force is $m_2g - T = m_2a$. Notice that the force of gravity is positive because it acts in the positive direction (downwards).

We now have a system of two equations with two unknowns ($T$ and $a$). Solving these equations will give us the acceleration of the system and the tension in the string. I know it might seem like a lot, but trust me, we're almost there! Taking the time to draw the free body diagrams and understand the forces acting on each mass is crucial to understanding the problem. The next step is to solve the system of equations. After solving the system of equations, we can calculate the numerical values for the acceleration and tension.

Solving for Acceleration and Tension

Alright, let's put on our math hats and solve for the acceleration ($a$) and the tension ($T$) in the string. We have two equations:

1. $T - m_1g = m_1a$
2. $m_2g - T = m_2a$

There are several ways to solve this system of equations. One common method is to solve for $T$ in one equation and substitute it into the other. Let's solve equation (1) for $T$:

$T = m_1a + m_1g$

Now, substitute this expression for $T$ into equation (2):

$m_2g - (m_1a + m_1g) = m_2a$

Simplify the equation:

$m_2g - m_1a - m_1g = m_2a$

Rearrange the equation to solve for $a$:

$m_2g - m_1g = m_1a + m_2a$

$g(m_2 - m_1) = a(m_1 + m_2)$

Finally, solve for $a$:

a = rac{g(m_2 - m_1)}{(m_1 + m_2)}

Now, we can plug in the values for $m_1$, $m_2$, and $g$:

a = rac{9.8 ext{ m/s}^2 (0.6 ext{ kg} - 0.2 ext{ kg})}{(0.2 ext{ kg} + 0.6 ext{ kg})}

a = rac{9.8 ext{ m/s}^2 (0.4 ext{ kg})}{0.8 ext{ kg}}

$a = 4.9 ext{ m/s}^2$

So, the acceleration of the system is $4.9 ext{ m/s}^2$. Now, let's solve for the tension $T$. We can use the equation $T = m_1a + m_1g$:

$T = (0.2 ext{ kg})(4.9 ext{ m/s}^2) + (0.2 ext{ kg})(9.8 ext{ m/s}^2)$

$T = 0.98 ext{ N} + 1.96 ext{ N}$

$T = 2.94 ext{ N}$

Therefore, the tension in the string is $2.94 ext{ N}$. Congratulations, we've solved the problem! We have found both the acceleration and the tension! The key is to take your time and follow these steps. Make sure to understand each step. Take your time, draw your free body diagrams, and keep your signs correct! Remember to perform this problem with different masses to help you learn and grasp the concept! This whole process has shown us how to approach these problems effectively.

Conclusion: Summary of Findings

So, to recap, guys, we've successfully tackled the problem of two masses hanging on a string! Let's summarize our findings:

• Acceleration of the system ($a$): The masses accelerate with a magnitude of $4.9 ext{ m/s}^2$. The heavier mass ($m_2$) accelerates downwards, and the lighter mass ($m_1$) accelerates upwards.
• Tension in the string ($T$): The tension in the string is $2.94 ext{ N}$. This is the force that the string exerts on each mass.

This problem highlights the beauty of physics – how a few fundamental laws can explain the motion of objects. We used Newton's Second Law and free-body diagrams to break down the problem into manageable steps. This approach is applicable to countless other physics problems. This problem is a fundamental one and it's a great example of the interplay between gravity, tension, and acceleration. This approach can be used on other problems as well, so knowing how to do this one is very important. Keep practicing, and you'll become a physics pro in no time!

And that's a wrap! Hope this explanation was helpful. If you have any questions, feel free to ask. Keep exploring the awesome world of physics!