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m₂ a. First, assume that you have lowered the friction on the plane to be essentially zero; give the kinetic and potential energies of the two particles before movement starts and just before the moment that particle 2 hits the ground. b. With what velocity is particle 2 hitting ground, and what is particle 1's uphill velocity at that moment? c. What is the maximum height reached by particle 1? d. Now, second, assume the lubricant only barely managed to get particle 1 unstuck; describe the ensuing motion in your own words. How much internal energy is generated by the motion?

m₂ a. First, assume that you have lowered the friction on the plane to be essentially zero; give the kinetic and potential energies of the two particles before movement starts and just before the moment that particle 2 hits the ground. b. With what velocity is particle 2 hitting ground, and what is particle 1's uphill velocity at that moment? c. What is the maximum height reached by particle 1? d. Now, second, assume the lubricant only barely managed to get particle 1 unstuck; describe the ensuing motion in your own words. How much internal energy is generated by the motion?

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**Energy Problem with Inclined Plane and Pulley System** Consider a system with two particles with masses \( m_1 \) and \( m_2 \), connected by a thin, lightweight rope passing over a lightweight and frictionless pulley. Particle \( m_1 \) is at the base of an inclined plane, while particle \( m_2 \) is at a height \( h \) equal to the total height of the ramp. Initially, there is no movement due to friction. You reduce friction by applying a lubricant, resulting in a friction coefficient \( \mu \) under \( m_1 \). **Diagram Explanation:** - The diagram shows an inclined plane with angle \( \alpha \). - Particle \( m_1 \) is positioned on this plane. - Particle \( m_2 \) is hanging vertically. - A pulley is located at the top of the incline. **Questions:** a. **Energy Calculation:** - Lower friction to near-zero. Determine kinetic and potential energies for both particles just before motion and before \( m_2 \) hits the ground. b. **Velocity Analysis:** - Calculate the velocity of \( m_2 \) upon hitting the ground and the corresponding uphill velocity of \( m_1 \). c. **Maximum Height:** - Determine the maximum height reached by particle \( m_1 \). d. **Internal Energy:** - Describe the motion assuming the lubricant only barely initiates it. Specify internal energy generated. e. **Numerical Solution:** - Given \( m_1 = 0.5 \, \text{kg},\, m_2 = 10 \, \text{kg},\, \alpha = 30^\circ,\, h = 10 \, \text{m},\, \mu = 1.0 \), calculate results for parts b, c, and d. This problem provides a practical scenario to apply energy principles, exploring dynamics involving friction and energy transformation in a mechanical system.
Transcribed Image Text:**Energy Problem with Inclined Plane and Pulley System** Consider a system with two particles with masses \( m_1 \) and \( m_2 \), connected by a thin, lightweight rope passing over a lightweight and frictionless pulley. Particle \( m_1 \) is at the base of an inclined plane, while particle \( m_2 \) is at a height \( h \) equal to the total height of the ramp. Initially, there is no movement due to friction. You reduce friction by applying a lubricant, resulting in a friction coefficient \( \mu \) under \( m_1 \). **Diagram Explanation:** - The diagram shows an inclined plane with angle \( \alpha \). - Particle \( m_1 \) is positioned on this plane. - Particle \( m_2 \) is hanging vertically. - A pulley is located at the top of the incline. **Questions:** a. **Energy Calculation:** - Lower friction to near-zero. Determine kinetic and potential energies for both particles just before motion and before \( m_2 \) hits the ground. b. **Velocity Analysis:** - Calculate the velocity of \( m_2 \) upon hitting the ground and the corresponding uphill velocity of \( m_1 \). c. **Maximum Height:** - Determine the maximum height reached by particle \( m_1 \). d. **Internal Energy:** - Describe the motion assuming the lubricant only barely initiates it. Specify internal energy generated. e. **Numerical Solution:** - Given \( m_1 = 0.5 \, \text{kg},\, m_2 = 10 \, \text{kg},\, \alpha = 30^\circ,\, h = 10 \, \text{m},\, \mu = 1.0 \), calculate results for parts b, c, and d. This problem provides a practical scenario to apply energy principles, exploring dynamics involving friction and energy transformation in a mechanical system.
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