Masses on incline system problem | Forces and Newton's laws of motion | Physics | Khan Academy
TLDRThe video script demonstrates a powerful method for analyzing a system of multiple objects by treating them as a single mass, simplifying complex problems. It uses the example of a 9kg mass hanging from a rope connected to a 4kg mass on a 30-degree incline with friction. By focusing on external forces and the total mass, the system's acceleration is calculated, ignoring internal forces. This approach first gives the system's acceleration and then allows for the determination of individual forces, such as tension, using Newton's second law for each mass separately.
Takeaways
- 🔍 The concept of treating multiple objects as a single mass simplifies complex problems by allowing us to find the system's acceleration without dealing with multiple equations and unknowns.
- 📈 In the given problem, a 9kg mass is hanging from a rope connected to a 4kg mass on a 30-degree incline with a coefficient of kinetic friction of 0.2.
- 🧠 The system's acceleration depends only on external forces and is found by dividing these forces by the total mass of the system, which in this case is 13kg (9kg + 4kg).
- 📊 The force of gravity on the 9kg mass is the driving force for the system's acceleration, calculated as 9 kg * 9.8 m/s² and considered positive.
- 🔧 The component of the 4kg mass's gravitational force that resists acceleration is calculated using 'm * g * sin(theta)', where theta is the incline angle.
- 🚫 The force of friction resisting the system's motion is the kinetic friction force, calculated as 'mu_k * Fn', where mu_k is 0.2 and Fn is the normal force.
- 🌐 The normal force on the incline is different from the total gravitational force due to the incline's angle and is calculated as 'm * g * cos(theta)'.
- ⚖️ The total mass cancels out internal forces, which allows us to neglect tension in the rope when calculating the system's acceleration.
- 🎯 Once the system's acceleration is known, individual forces such as tension can be calculated using Newton's second law for the 9kg mass alone.
- 🔄 The process of treating a multi-mass system as a single entity and then analyzing individual masses is a two-step approach that provides a quick solution for complex problems.
- 📌 The final calculated acceleration for the system is 4.75 m/s², and the tension in the rope is found to be 45.5 newtons.
Q & A
What is the main concept explained in the script?
-The main concept explained in the script is treating multiple objects as a single mass to find the system's acceleration and then using that to determine individual forces or motions within the system.
What are the two masses mentioned in the script and how are they connected?
-The two masses mentioned in the script are a 9 kg mass hanging from a rope and a 4 kg mass sitting on an incline. The rope passes over a pulley and is connected to the 4 kg mass on the incline.
What is the angle of the incline in the problem?
-The angle of the incline in the problem is 30 degrees.
What is the coefficient of kinetic friction between the incline and the 4 kg mass?
-The coefficient of kinetic friction between the incline and the 4 kg mass is 0.2.
How does the system's acceleration depend on external forces?
-The system's acceleration depends on external forces as it only considers the net external force acting on the system and divides it by the total mass of the system, neglecting internal forces.
What is the direction defined as positive in the script?
-The direction defined as positive in the script is the direction of motion of the 4 kg mass up the incline, which is also the direction of the acceleration of the system.
How is the component of gravity acting on the 4 kg mass on the incline calculated?
-The component of gravity acting on the 4 kg mass on the incline is calculated using the formula 'm * g * sin(theta)', where m is the mass, g is the acceleration due to gravity, and theta is the angle of the incline.
What is the normal force on the 4 kg mass on the incline and how does it relate to friction?
-The normal force on the 4 kg mass on the incline is equal to the mass times the acceleration due to gravity times the cosine of the incline's angle (m * g * cos(theta)). It is important for calculating friction since the force of kinetic friction is the product of the coefficient of kinetic friction (mu_k) and the normal force (Fn).
What is the calculated acceleration of the system?
-The calculated acceleration of the system is 4.75 meters per second squared.
How can you find the tension in the rope once you have the system's acceleration?
-Once you have the system's acceleration, you can find the tension in the rope by considering the 9 kg mass individually, applying Newton's second law, and solving for the tension, which turns out to be 45.5 newtons.
Why is the tension in the rope less than the force of gravity on the 9 kg block?
-The tension in the rope is less than the force of gravity on the 9 kg block because the block is accelerating downwards, indicating that the net force is directed downwards, which means the tension must be less than the gravitational force to allow for this acceleration.
Outlines
📚 Introduction to System Dynamics
This paragraph introduces the concept of analyzing a system of multiple objects by treating them as a single mass. It presents a challenging problem involving a 9kg mass hanging from a rope, connected to a 4kg mass on a 30-degree incline with a coefficient of kinetic friction of 0.2. The paragraph explains that solving this problem using Newton's second law for each mass separately would be complex, involving multiple equations with multiple unknowns. Instead, the paragraph advocates for a system approach, where the acceleration of the entire system can be determined by considering only the external forces and dividing by the total mass of the system.
🔍 Calculating System Acceleration and Internal Forces
The second paragraph delves into the process of calculating the system's acceleration by identifying the forces acting on the masses. It explains how to determine the net force by considering the gravitational force on the 9kg mass, the component of the 4kg mass's weight along the incline, and the frictional force based on the normal force and the coefficient of kinetic friction. The paragraph clarifies that internal forces, such as tension in the rope, cancel each other out and do not need to be considered when treating the system as a single mass. It then demonstrates how to calculate the acceleration of the system, which is found to be 4.75 m/s². The paragraph also discusses how to find the internal force of tension by analyzing the 9kg mass individually after determining the system's acceleration.
Mindmap
Keywords
💡multiple objects
💡single mass
💡acceleration
💡Newton's second law
💡external forces
💡internal forces
💡kinetic friction
💡angle of incline
💡tension
💡components of force
💡net force
Highlights
The concept of treating multiple objects as a single mass is introduced as a powerful approach to simplify complex problems.
A 9kg mass hanging from a rope connected to a 4kg mass on a 30-degree incline is used as an example to illustrate this method.
The coefficient of kinetic friction between the incline and the 4kg mass is given as 0.2, adding a realistic aspect of resistance to the problem.
The challenge of solving the problem using traditional Newton's second law for multiple boxes with multiple unknowns is acknowledged.
The system's acceleration is determined by external forces and the total mass, ignoring the internal forces which cancel each other out.
The direction of motion is defined as positive, which is crucial for applying the method to problems involving angles and inclined planes.
The force of gravity on the 9kg mass is identified as the driving force for the system's acceleration.
A detailed explanation of how to calculate the component of gravity resisting the motion along the incline is provided.
The force of friction is calculated using the force of kinetic friction and the normal force on the incline.
The total mass of the system is used to find the combined acceleration, which is 4.75 meters per second squared.
The method allows for the quick determination of the system's acceleration without needing to calculate individual internal forces.
Once the system's acceleration is known, individual forces such as tension can be calculated for each mass using Newton's second law.
The example demonstrates that the tension in the rope is less than the force of gravity on the 9kg block due to the net force pointing downward.
The process of switching from treating the two-mass system as a single object to analyzing individual masses is explained.
The final calculated tension is 45.5 newtons, showcasing the practical application of the method in finding specific forces within the system.
The transcript emphasizes the efficiency of this method in quickly obtaining acceleration and subsequently analyzing individual forces within a system of masses.
Transcripts
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