Introduction to Pulley Systems: Solving for Acceleration Using Newton's Laws and Euler-Lagrange Equations

Understanding the dynamics of a simple pulley system can be a challenging yet fascinating task. In this article, we will delve into the mechanics of such a system when one mass is subjected to an external force. We will employ two distinct methods: Newton's second law and the Euler-Lagrange equation to derive the acceleration of the masses involved.

Newton's Second Law: A Step-by-Step Analysis

Consider the scenario where two masses, each of 60 grams, are attached to a frictionless pulley. When an additional 10 grams are added to one of the masses, it becomes 70 grams. Our objective is to determine the downward acceleration of the 70-gram mass. This can be solved using Newton's second law of motion: Fma.

Initial Setup

m1 60 grams hanging on one side. m2 60 grams hanging on the other side. When 10 grams are added to m1, it becomes m1 70 grams.

Net Forces

Forces acting on each mass are due to gravity:

F1 m1g 70 grams * 980 cm/s2 68600 dyn F2 m2g 60 grams * 980 cm/s2 58800 dyn

Here, g is the acceleration due to gravity, approximately 980 cm/s2.

Calculating Forces and Net Force

The net force Fnet acting on the system:

Fnet  F1 - F2 68600 dyn - 58800 dyn 9800 dyn

Total Mass and Acceleration

The total mass mtotal of the system:

mtotal  m1   m2 70 grams   60 grams 130 grams

Using Newton's second law: F ma, we can find the acceleration a:

a  Fnet / mtotal 9800 dyn / 130 grams≈ 75.38 cm/s2

Thus, the downward acceleration of the mass with 70 grams is approximately 75.38 cm/s2.

Applying the Euler-Lagrange Equation: A Different Approach

For a more advanced approach, let's use the Euler-Lagrange equations. The total kinetic energy (KE) and total potential energy (PE) for the system can be expressed as:

Total Kinetic Energy (KE):

[ KE frac{1}{2} cdot 60 cdot dot{x}^2 frac{1}{2} cdot 70 cdot dot{x}^2 65 dot{x}^2 ]

Total Potential Energy (PE):

[ PE 60 cdot 980 cdot x 70 cdot 980 cdot (10 - x) 5880 686000 - 6860 686000 - 980 ]

The Lagrangian L is given by:

[L KE - PE frac{1}{2} cdot 65 cdot dot{x}^2 - (686000 - 980)]

The Euler-Lagrange equation is:

[frac{d}{dt} left( frac{partial L}{partial dot{x}} right) frac{partial L}{partial x}]

Deriving the Euler-Lagrange equation for the Lagrangian L:

[frac{d}{dt} (130 dot{x}) 9800][130 ddot{x} 9800][ddot{x} frac{9800}{130} approx 75.38 text{cm/s}^2]

This confirms the earlier calculation using Newton's second law.

Conclusion

Both methods, Newton's second law and the Euler-Lagrange equatino, yield the same result, verifying the accuracy of both approaches. This illustrates the robustness of physical laws and the elegance of different mathematical frameworks in solving practical problems in mechanics.