= 1. As shown in the figure m₁ 1.4kg, m₂ = 5.4kg and m3 = 8.6kg. Masses 1 and 3 are on the ends of a weightless rope which does not slip as it goes over the disk. Mass 2 is a solid disk of radius R = 36cm. The moment of inertia is I₂ = 2m₂R². Mass 1 is initially resting on the floor and mass 3 is suspended at a height h= 1.4m. When released mass 3 falls toward the floor. Using conservation of energy determine the velocity of the system just before m3 hits the floor. What is the angular velocity of the disk? What is the kinetic energy of each mass at this time? m₂
= 1. As shown in the figure m₁ 1.4kg, m₂ = 5.4kg and m3 = 8.6kg. Masses 1 and 3 are on the ends of a weightless rope which does not slip as it goes over the disk. Mass 2 is a solid disk of radius R = 36cm. The moment of inertia is I₂ = 2m₂R². Mass 1 is initially resting on the floor and mass 3 is suspended at a height h= 1.4m. When released mass 3 falls toward the floor. Using conservation of energy determine the velocity of the system just before m3 hits the floor. What is the angular velocity of the disk? What is the kinetic energy of each mass at this time? m₂
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Transcribed Image Text:### Physics Problem on Conservation of Energy
**Problem Statement:**
As shown in the figure:
- \( m_1 = 1.4 \, \text{kg} \)
- \( m_2 = 5.4 \, \text{kg} \)
- \( m_3 = 8.6 \, \text{kg} \)
Masses 1 and 3 are on the ends of a weightless rope which does not slip as it goes over the disk. Mass 2 is a solid disk of radius \( R = 36 \, \text{cm} \). The moment of inertia of the disk is \( I_2 = \frac{1}{2} m_2 R^2 \).
- Mass 1 is initially resting on the floor.
- Mass 3 is suspended at a height \( h = 1.4 \, \text{m} \).
When released, mass 3 falls toward the floor. Using the conservation of energy, determine:
1. The velocity of the system just before \( m_3 \) hits the floor.
2. The angular velocity of the disk.
3. The kinetic energy of each mass at this time.
**Explanation and Calculations:**
1. **Determine the Velocity of the System:**
To find the velocity of the masses and disk, apply the conservation of mechanical energy. Initially, only mass 3 has gravitational potential energy (since it is at height \( h \)) and there is no kinetic energy. Just before mass 3 hits the floor, all of this potential energy would be converted into kinetic energy of masses 1, 2 (disk), and 3.
2. **Calculate the Angular Velocity of the Disk:**
Use the relationship between linear velocity \( v \) and angular velocity \( \omega \) of the disk: \( v = \omega R \).
3. **Find the Kinetic Energy of Each Mass:**
Calculate the kinetic energy of each mass using the formula:
- For translational motion: \( KE_{\text{translational}} = \frac{1}{2} m v^2 \)
- For rotational motion of the disk: \( KE_{\text{rotational}} = \frac{1}{2} I \omega^2 \)
These steps will allow solving for the corresponding velocities and kinetic energies based on the principles of conservation of energy.
Transcribed Image Text:This diagram depicts a classical physics problem involving three masses (\(m_1\), \(m_2\), and \(m_3\)), a pulley of radius \(R\), and a falling distance \(l\). Below is an explanation and transcription of the diagram suitable for an educational website:
### Description
In the diagram, a system comprising three masses (\(m_1\), \(m_2\), and \(m_3\)) is depicted. The masses are connected by a rope over a pulley.
### Detailed Explanation
1. **Mass \(m_1\)**: This mass (\(m_1\)) lies on a horizontal surface.
2. **Mass \(m_2\)**: This is the mass attached to the rope which passes over a pulley. The pulley has a radius \(R\).
3. **Mass \(m_3\)**: This mass (\(m_3\)) is suspended in the air and can move vertically.
The system is typically used to study the principles of mechanics, including forces and motion when different masses are connected over a pulley.
1. **Pulley and Radius \(R\)**:
- The pulley is depicted as a circle with a connecting radius \(R\). This radius is crucial for calculating the torque and moment of inertia involving the pulley system.
2. **Vertical Movement and Distance \(l\)**:
- \(l\) is the vertical distance traveled by the mass \(m_3\). This distance is measured from a reference point.
### Transcription of Labels
- \(R\): Radius of the pulley.
- \(m_1\): Mass 1 placed on the ground level.
- \(m_2\): Mass 2 which is on the pulley.
- \(m_3\): Mass 3 which hangs vertically.
- \(l\): Vertical distance.
This setup illustrates the interplay of gravitational forces, tension in the rope, and rotational motion in a pulley system. Calculations often involve Newton's Second Law, equations of motion, and energy conservation principles to analyze the movement and forces within this system.
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