Lagrange Equation

Abstract

In this chapter, we will learn the knowledge on Lagrange equation. We mainly learn how to use it to solve problems in dynamics, and some examples are given.

Exercises

• 14.1 As shown in Fig. 14.4, two wheels and one weight are linked by one rope. The wheel A has mass m, and its radius is r, which is in pure rolling on the oblique surface with an angle $\beta$. The mass center of the wheel A is linked with a weight with mass 2m, and the rope is inextensible. The fixed wheel C has the radius r, and its mass is ignored. Please calculate the following variables:

  Fig. 14.4

  

  Two wheels and one weight linked together

  • (1) The acceleration a_A of the wheel center A.

  • (2) The tension T₁ of the rope ①.

  • (3) The frictional force F between the wheel A and the oblique surface.

• 14.2 As shown in Fig. 14.5, two wheels and one weight are linked together. The gravity of the weight A is P, and its dynamic frictional coefficient with the oblique surface is $f^{\prime}$. The angle between the oblique surface and the horizontal surface is $\alpha$, and the weight of the wheel B is P with radius R. There is no relative sliding between the rope and the wheel. The circular disk C is in pure rolling, with the gravity P and radius r. The two segments of the rope are parallel the oblique surface and horizontal surface, respectively. When the weight A falls down along the oblique surface from the stationary state with the distance s, please calculate the following parameters:

  Fig. 14.5

  

  Two wheels and one weight linked together

  • (1) The angular velocity and angular acceleration of the wheel B.

  • (2) The tension force between the weight A and the wheel B.

Answers

• 14.1 (1) $a_{A} = \frac{ 4g - 2g\sin \beta }{7}$ (2) $T_{1} = \frac{ 6mg + 4mg\sin \beta }{7}$ (3) $f = \frac{{\left( { 2g - g\sin \beta } \right)m}}{14}$

• 14.2 (1) $\omega_{B} = \frac{{\sqrt {6sg(\sin \alpha - f^{\prime}\cos \alpha )} }}{3R}$, $\varepsilon_{B} = \frac{{g(\sin \alpha - f^{\prime}\cos \alpha )}}{3R}$

  (2) $F_{AB} = \frac{{2P\left( {\sin \alpha - f^{\prime}\cos \alpha } \right)}}{3}$.