# When Will the Rod Fall without Slipping?

• zemansky
The Lagrangian is a function that describes the dynamics of a system in terms of its generalized coordinates and their time derivatives. In this problem, the Lagrangian would be used to derive the equation of motion and the mechanical energy of the system. From there, we can use Newton's second law or D'Alembert's principle to find the normal and friction forces acting on the rod. Finally, we need to determine the condition for the rod to fall all the way to the ground without slipping.
zemansky

## Homework Statement

A uniform rod of mass M and length L is placed vertically on a flat surface. The coefficient of friction between the rod and the surface is μ. The rod is beginning to fall, initially without slipping. To describe the motion of the rod during this stage:
(a) Write down the Lagrangian and derive the equation of motion for the generalized coordinate
$\phi$.

b) From the Lagrangian derive the expression for the mechanical en- ergy of the system and write down the energy conservation law.

(c) Use Newton’s second law (or D’Alembert’s principle) to derive expressions for the normal (N) and friction (f) forces acting on the rod. Express the answers in terms of $\phi$ only. You can use the results of parts (a) and (b) for this step.

(d) Under what condition will the rod fall all the way to the ground without slipping?

## The Attempt at a Solution

No attempt to solve this problem has been done. I need help to set up the problem!

Last edited:
Hi zemansky, welcome to PF!

You need to show what you know. What is "Lagrangian"?

ehild

## What is the Falling Rod Problem?

The Falling Rod Problem is a physics problem that involves a rod balanced on a pivot point and then released to fall under the influence of gravity. The question is, what will the motion of the rod be as it falls?

## What factors affect the motion of the falling rod?

The motion of the falling rod is affected by several factors, including the length and mass of the rod, the distance between the pivot point and the center of mass, and the initial angle at which the rod is released.

## What is the equation for the motion of the falling rod?

The equation for the motion of the falling rod is known as the Euler-Lagrange equation. It takes into account the conservation of energy and the dynamics of the system, and can be solved using calculus to determine the motion of the rod.

## What is the significance of the Falling Rod Problem?

The Falling Rod Problem is significant because it illustrates the principles of classical mechanics and the importance of conservation of energy and momentum. It also has real-world applications, such as in understanding the motion of objects in freefall or the stability of structures.

## Are there any real-life examples of the Falling Rod Problem?

Yes, there are several real-life examples of the Falling Rod Problem, such as a falling building or a falling tree. It can also be applied to sports, such as pole vaulting or high jump, where the athlete's body can be modeled as a falling rod.

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