Two masses are attached to a pulley and released from rest, with M2 on the ground. M1 begins to fall and M2 rises. There is friction in the axle that produces a constant torque of t Nm on the rotating object. Find the speed of M1 just before it hits the ground in terms of the given information (M1, M2, M,, R, T, H, t, and our usual constants) Two solid cylinders are glued together to form a pulley,
Kinematics
A machine is a device that accepts energy in some available form and utilizes it to do a type of work. Energy, work, or power has to be transferred from one mechanical part to another to run a machine. While the transfer of energy between two machine parts, those two parts experience a relative motion with each other. Studying such relative motions is termed kinematics.
Kinetic Energy and Work-Energy Theorem
In physics, work is the product of the net force in direction of the displacement and the magnitude of this displacement or it can also be defined as the energy transfer of an object when it is moved for a distance due to the forces acting on it in the direction of displacement and perpendicular to the displacement which is called the normal force. Energy is the capacity of any object doing work. The SI unit of work is joule and energy is Joule. This principle follows the second law of Newton's law of motion where the net force causes the acceleration of an object. The force of gravity which is downward force and the normal force acting on an object which is perpendicular to the object are equal in magnitude but opposite to the direction, so while determining the net force, these two components cancel out. The net force is the horizontal component of the force and in our explanation, we consider everything as frictionless surface since friction should also be calculated while called the work-energy component of the object. The two most basics of energy classification are potential energy and kinetic energy. There are various kinds of kinetic energy like chemical, mechanical, thermal, nuclear, electrical, radiant energy, and so on. The work is done when there is a change in energy and it mainly depends on the application of force and movement of the object. Let us say how much work is needed to lift a 5kg ball 5m high. Work is mathematically represented as Force ×Displacement. So it will be 5kg times the gravitational constant on earth and the distance moved by the object. Wnet=Fnet times Displacement.
(Physics Problem)
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![**Title: Mechanics Problem Involving Pulley and Masses**
**Introduction:**
This example explores the dynamics of a system where two masses are connected to a pulley and are released from a state of rest. The system includes friction in the axle, which results in a constant torque being applied to the rotating object. We aim to determine the speed of mass M1 just before it impacts the ground.
**Problem Statement:**
Two masses are attached to a pulley and released from rest, with mass M2 on the ground. As the system evolves, M1 begins to fall while M2 rises. There is friction in the axle that produces a constant torque \( \tau \) Nm on the rotating object. Determine the speed of M1 just before it hits the ground in terms of the given variables (M1, M2, \(M_p\), \(R\), \(T\), \(H\), \( \tau \), and our typical physical constants).
### Components of the System:
1. **Masses and Pulley:**
- Two masses: M1 and M2
- A complex pulley composed of two solid cylinders:
- **Cylinder 1:** Radius \( R \) and mass \( M_p \), thickness \( T \)
- **Cylinder 2:** Radius \( 2R \) and mass \( 2M_p \), thickness \( 2T \)
2. **Diagram Description:**
- **Pulley System (Left Side):** Two masses (M1 and M2) are connected to a pulley, with M2 on the ground and M1 at an elevation \( H \).
- **Pulley Description (Right Side):** The pulley consists of two solid cylinders glued together:
- Cylinder 1 has a radius \( R \) and a mass \( M_p \).
- Cylinder 2 has a radius \( 2R \) and a mass \( 2M_p \).
- Thickness specifications for the cylinders are \( T \) for Cylinder 1 and \( 2T \) for Cylinder 2.
### Diagram Analysis:
- **Left Diagram:**
- The pulley is shown with both masses M1 (falling) and M2 (rising).
- Both masses are attached via strings that pass through the pulley's edge.
- The height \( H \) is indicated as the initial vertical distance between M1 and the ground.
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