Lagrange equations: Two blocks and a string

In summary, the kinetic energy of the string is given by the formula ##T=\frac{1}{2}mv^2##, where ##m## is the mass of the whole string and ##v## is the speed of the string. This formula takes into account the contribution of the string in both the vertical and horizontal directions, as the string is in motion in both directions. However, in the given problem, only the vertical motion of the string is considered, leading to a formula of ##T_{string}=\frac{1}{2}m\dot{y}##, which neglects the contribution of the string's horizontal motion. To account for this, the correct formula for the kinetic energy of the string should be
  • #1
Zamarripa
11
0
Homework Statement
Two blocks, each of mass M, are connected by an extensionless, uniform string of length l. One block is placed on a smooth horizontal surface, and the other block hangs over the side, the string passing over a frictionless pulley. Describe the motion of the system in two cases, first when the mass of the string is negligible, and second when the string has a mass m.
Relevant Equations
$$T_{string}=\frac{1}{2}m\dot{y} $$
I've problems understanding why the kinetic energy of the string is only

Captura de Pantalla 2020-05-29 a la(s) 14.37.32.png

$$T_{string}=\frac{1}{2}m\dot{y} $$

Why the contribution of the string in the horizontal line isn't considered?
 
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  • #2
What's the formula for KE?
 
  • #3
PeroK said:
What's the formula for KE?
$$T=\frac{1}{2}mv^2$$
 
  • #4
Zamarripa said:
$$T=\frac{1}{2}mv^2$$
Which is not what you're using?
 
  • #5
Yes, but what about the contribution on the horizontal?
 
  • #6
Zamarripa said:
Yes, but what about the contribution on the horizontal?

What contribution from the horizontal?

The problem is that you have ##\frac 1 2 m \dot y## instead of ##\frac 1 2 m \dot y^2##.
 
  • #7
PeroK said:
What contribution from the horizontal?

The problem is that you have ##\frac 1 2 m \dot y## instead of ##\frac 1 2 m \dot y^2##.
The string at the horizontal is also moving
 
  • #8
Zamarripa said:
The string at the horizontal is also moving
So, what do you think the KE of the string should be?
 
  • #9
PeroK said:
So, what do you think the KE of the string should be?
$$\frac{1}{2}m\frac{(s-x)}{l}\dot{x}$$

where ##m\frac{(s-x)}{l}## is the mass of the string in the horizontal
 
  • #10
Zamarripa said:
$$\frac{1}{2}m\frac{(s-x)}{l}\dot{x}$$

where ##m\frac{(s-x)}{l}## is the mass of the string in the horizontal

1) The whole string is moving, so you need the whole mass of the string: ##m##.

2) ##\dot x = \dot y##, as the string is inextensible.

3) You need to use ##v^2 = \dot x^2 = \dot y^2## for KE. You can't use ##\dot x## or ##\dot y##.
 
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Related to Lagrange equations: Two blocks and a string

1. What are Lagrange equations?

Lagrange equations are a set of equations used in classical mechanics to describe the motion of a system. They are derived from the principle of least action and are used to find the equations of motion for a system in terms of generalized coordinates.

2. How are Lagrange equations applied to a system with two blocks and a string?

In the case of a system with two blocks connected by a string, Lagrange equations can be used to find the equations of motion for each block in terms of the position, velocity, and acceleration of the blocks and the tension in the string.

3. What is the significance of the Lagrange equations in physics?

The Lagrange equations are significant in physics because they provide a powerful and elegant method for solving problems in classical mechanics. They allow for the description of complex systems and can be used to find the equations of motion for any system that can be described in terms of generalized coordinates.

4. How do Lagrange equations differ from Newton's laws of motion?

While Newton's laws of motion describe the relationship between the forces acting on a system and its resulting motion, Lagrange equations take a more abstract approach by describing the motion of a system in terms of generalized coordinates. This allows for a more flexible and comprehensive analysis of complex systems.

5. Are Lagrange equations applicable to systems other than two blocks and a string?

Yes, Lagrange equations can be applied to any system that can be described in terms of generalized coordinates, such as systems with multiple particles, rigid bodies, or even systems with constraints. They are a powerful tool in classical mechanics and can be used to solve a wide range of problems.

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