Solving Pulley Systems: Kinetic Energy and Lagrangian

In summary, the conversation revolves around a question about calculating the kinetic energy of a system involving pulleys and masses using methods of generalized coordinates and Lagrangian functions. The main challenge is finding the relations between the different coordinates and using a basic formula for kinetic energy. A hint is given to help with solving the problem.
  • #1
tom777
3
0
Hello guys!
Another question appeared during the preparation for my exam .
For some reason I don't feel 100% comfortable when dealing with
systems including pulleys and masses.
In the given exercise I'm supposed to calculate the kinetic energy T of the system
(which ultimately leads to the Lagrangian)

Homework Statement


Scan of the problem:
http://img138.imageshack.us/img138/2064/question2.png [Broken]

Homework Equations


The problem is solved by using methods of generalized coordinates
and Lagrangian functions.

The Attempt at a Solution


So the thing is...I sort of know where most of the terms come from.
For example the (I_2/ (R_2)^2 ) probably comes from (d/dt y_{1} ) = (d/dt \theta_{2}) * R_1. However I'm lacking a concrete, rigorous approach to the problem.
What I mean by that is:
Say in cartesian coordinates the formula for the kinetic energy is given by:
T = 0.5 * m_1 * y_{mass 1} + 0.5 * m_2 * y_{mass 2}
How do I rigorously solve this problem now? I'm sort of puzzled - especially about
the last term in the equation for T in the scan. It seems as though some kind
of binomial formula might have been applied. But I'm not sure though.

I'd be really really happy if you could help me or give me a hint
on home to tackle these pulley-problems.

Thanks a lot in advance!
 
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  • #2
(d/dt y_{1} ) = (d/dt \theta_{2}) * R_1
Uhm... not right :frown:

The most challenging thing in this problem is to find the relations of [tex]\dot{\theta _1}, \dot{\theta _2} , \dot{y _1} , \dot{y _2}[/tex]. I suggest you sit down and write down the relations without thinking about the final result of T, then write down T in its basic form, i.e. [tex]T_{linear}=\frac{1}{2}m\dot{x}^2[/tex] and [tex]T_{rotate}=\frac{1}{2}I\dot{\phi}^2[/tex].

Hint: There are 2 equations relating the coordinates. The 1st one contains [tex]\dot{\theta _1}[/tex] and [tex]\dot{y _2}[/tex] only. The 2nd one contains [tex]\dot{\theta _2} , \dot{y _1} , \dot{y _2}[/tex]. The radii are included, of course.
 

1. What is a pulley system and how does it work?

A pulley system is a mechanical device that uses a wheel with a groove and a rope or belt to change the direction of a force. It works by distributing the force of an object's weight over multiple ropes, making it easier to lift or move the object.

2. How is kinetic energy related to pulley systems?

Kinetic energy is the energy an object possesses due to its motion. In pulley systems, kinetic energy is transferred from the object being lifted to the pulley itself and then to the rope. As the rope moves, it converts the kinetic energy into mechanical work to lift the object.

3. What is the Lagrangian of a pulley system?

The Lagrangian of a pulley system is a mathematical function that describes the system's energy, taking into account both kinetic and potential energy. It is used in the Lagrangian mechanics approach to solve problems involving motion and energy in mechanical systems.

4. How do you calculate the mechanical advantage of a pulley system?

The mechanical advantage of a pulley system is the ratio of the output force to the input force. To calculate it, divide the load force (the weight of the object being lifted) by the effort force (the force applied to the rope). The mechanical advantage can also be determined by counting the number of ropes supporting the object and subtracting one.

5. What are the common applications of pulley systems?

Pulley systems have a wide range of applications, including lifting heavy objects, moving loads horizontally, and changing the direction of a force. They are commonly used in cranes, elevators, and simple machines such as well pumps. They can also be found in everyday objects such as window blinds and exercise equipment.

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