'AP Physics 2023 Exam Solutions|Q1. PART C ( i ) | Complete Step-by-Step Answers and Explanations"
TLDRThis script discusses the conservation of total energy in a simple harmonic motion (SHM) system. Initially, a single block undergoes SHM with a spring. When a second block is gently placed on the first, they stick together and continue SHM with a combined mass. Despite the change in mass, the total energy remains constant, as no external work is done. The kinetic and potential energy graph remains the same, with the maximum energy still being four joules. The amplitude and total energy are preserved, but the speed at the mean position decreases due to increased mass. The script illustrates the physics principle of energy conservation in SHM.
Takeaways
- 🔍 The question is about explaining why the graphs of kinetic energy and potential energy remain the same after a block is added to the system.
- 🔄 Initially, a single block was performing simple harmonic motion (SHM) with a spring.
- 📉 A 3M mass block is gently placed on the initial block, and they stick together, continuing SHM with a combined mass of 4M.
- ⚡ The total energy of the system remains constant at 4 joules, both before and after adding the new block.
- 🎯 The maximum kinetic energy and maximum potential energy remain at 4 joules each.
- 🔗 The internal forces between the two blocks do not perform any work, hence no external energy is added or lost.
- 🐌 The speed at the mean position is reduced due to the increased mass of the system.
- 📐 The total energy of the system is a sum of kinetic and potential energies at any position.
- 📏 The amplitude of the SHM remains unchanged, maintaining the same energy distribution.
- 📊 The graphical representation of kinetic and potential energies still shows a maximum of 4 joules each, leading to the same energy curves as before.
Q & A
What is the main topic of the video script?
-The main topic of the video script is the conservation of total energy in a system involving simple harmonic motion (SHM) with two blocks of different masses.
What is simple harmonic motion (SHM)?
-Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction, typically described by the equation F = -kx.
Why do the two graphs of kinetic energy versus potential energy remain the same after the second block is added?
-The two graphs remain the same because the total energy of the system is conserved. No external work is done on the system, and there is no loss of energy when the second block is gently placed on the first block.
What happens to the amplitude of the SHM when the second block is added?
-The amplitude of the SHM remains the same because the total energy of the system, which determines the amplitude, is conserved.
How is the speed of the combined system at the mean position affected by the addition of the second block?
-The speed of the combined system at the mean position is slower due to the increased mass, even though the total energy and amplitude remain the same.
What is the significance of the internal forces between the two blocks in terms of energy conservation?
-The internal forces between the two blocks, which cause them to stick together, do not do any work on the system. This means they do not contribute to a change in the total energy of the system.
What is the relationship between kinetic energy, potential energy, and total energy in the context of this script?
-In the context of this script, the total energy of the system is the sum of kinetic and potential energy at any given point in time. The total energy remains constant throughout the motion.
What is the maximum kinetic energy of the system before and after the second block is added?
-The maximum kinetic energy of the system remains the same before and after the second block is added, which is 4 joules, assuming that was the initial total energy of the system.
How does the mass of the system change when the second block is added?
-The mass of the system changes from the initial mass 'M' to a combined mass of '4M' when the second block of 3M is added and they stick together.
What is the formula for the maximum potential energy stored in the spring in the system?
-The formula for the maximum potential energy stored in the spring is (1/2)kx^2, where 'k' is the spring constant and 'x' is the displacement from the equilibrium position.
What does the dotted curve in the graph represent?
-The dotted curve in the graph represents the relationship between kinetic and potential energy over the course of the SHM, showing that the total energy is conserved and remains constant.
Outlines
🔁 Conservation of Energy in Simple Harmonic Motion
The first paragraph discusses the conservation of energy principle in the context of a simple harmonic motion (SHM) system. Initially, a single block with mass M undergoes SHM with a spring. When a second block of mass 3M is gently placed on the first, they stick together and continue to perform SHM with a combined mass of 4M. Despite the change in mass, the total energy of the system remains constant at 4 joules. The paragraph explains that the maximum kinetic and potential energy values remain the same, but the speed of the blocks at the mean position decreases due to the increased mass. The internal forces between the blocks do not do any work, and no external energy is added or lost, resulting in the same energy graph for kinetic and potential energy as before.
📊 Graph Analysis of Kinetic and Potential Energy
The second paragraph continues the discussion on energy conservation but focuses on the graphical representation of kinetic and potential energy in the SHM system. It explains that even after the addition of the second block, the amplitude of the SHM remains unchanged, and the maximum potential energy is still equal to the total energy of the system, which is 4 joules. The paragraph further clarifies that the speed at the mean position is reduced due to the increased mass, but the total energy at any point in the motion is the sum of kinetic and potential energy, which remains constant. The graph would show the same pattern as before, with maximum potential energy at the extreme positions and maximum kinetic energy at the mean position, confirming the conservation of energy principle.
Mindmap
Keywords
💡Simple Harmonic Motion (SHM)
💡Potential Energy
💡Kinetic Energy
💡Total Energy
💡Energy Conservation
💡Amplitude
💡Mass
💡Velocity
💡Spring Constant (k)
💡Mean Position
Highlights
The two graphs must be the same because the total energy of the system remains unchanged after placing the 3M block on the M block.
No external work is done when the 3M block is gently placed on the M block, so the total energy is conserved.
The internal force between the two blocks does not do any work on the system, maintaining the total energy.
The maximum kinetic and potential energy remain the same at 4 joules after the blocks are combined.
The amplitude of the simple harmonic motion (SHM) does not change, remaining from 0 to L and 0 to -L.
The speed at the mean position will be slower due to the increased mass after combining the blocks.
The kinetic energy at any given point is half the mass times the velocity squared.
The potential energy at any position X is given by half the spring constant times the displacement squared.
The total energy at any position X is the sum of kinetic and potential energy, which remains constant.
The maximum potential energy of the system is still equal to the total energy of 4 joules.
The speed at the mean position for the 4M block is determined by the total energy divided by the new mass.
The graph of kinetic and potential energy will show the same maximum values as before, with kinetic energy at zero displacement and potential energy at maximum displacement.
The dotted curve in the graph represents the relationship between kinetic and potential energy, which remains the same after the blocks are combined.
The physics principle of energy conservation explains why the graphs of kinetic and potential energy are the same before and after the blocks are combined.
The system's total energy before and after the blocks are combined is 4 joules, indicating no energy loss or gain.
The forces between the blocks that keep them together do not perform work, preserving the system's total energy.
The potential energy versus kinetic energy graph remains consistent with the conservation of energy principle in physics.
Transcripts
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