# Variable mass problem (integration help)

• Liquidxlax
In summary, the conversation discusses solving a problem involving a uniform chain of length L and density /rho(kg/m), initially stationary on a horizontal, frictionless table. The question is how much time passes before the entire chain leaves the table. The attempt at a solution involves an integral and the use of the equation arccosh(x) = log(sqrt(x2-1)+x). The discrepancy in the answer is resolved by realizing that l/g should be square rooted for the correct dimension of time.

## Homework Statement

I've pretty much solved it, but I'm unsure of my final integration

A uniform chain of length L and density /rho(kg/m) is initially stationary on a horizontal, frictionless table, with part of the chain (length yo) hanging over the edge. How much time passes before the entire chain has left the table?

## Homework Equations

arccosh(x) = log(sqrt(x2-1)+x)

## The Attempt at a Solution

I don't think i need to put all the work I've done.

my integral

$$\int \sqrt{(y^{2}-y^{2}_{o})g/l}^{-1/2}$$

$$arccosh( \sqrt{l/g}*y/y_{o})$$

or

$$\sqrt{l/g}*log(2(\sqrt{y^{2}-y^{2}_{o}}+y))$$

yet on wolfram and other websites they say that l/g should not be square rooted. Yet i don't see why.

thanks

L/g has to be square rooted. If not, the dimension of the time is wrong.

ehild

ah yes, thank you. I should have noticed that

## 1. What is the variable mass problem?

The variable mass problem is a mathematical concept in which the mass of an object changes over time. This can occur due to the addition or removal of mass, or due to the redistribution of mass within the object.

## 2. How is the variable mass problem relevant in science?

The variable mass problem is relevant in various fields of science, such as physics, chemistry, and engineering. It helps to understand the motion and behavior of objects that have changing mass, such as rockets, airplanes, and chemical reactions.

## 3. What is the equation for solving the variable mass problem?

The equation for solving the variable mass problem is known as the "rocket equation" and is given by:
Δv = ve * ln (m0 / mf)
Where Δv is the change in velocity, ve is the exhaust velocity, m0 is the initial mass, and mf is the final mass.

## 4. Can you provide an example of the variable mass problem in action?

One example of the variable mass problem is the motion of a rocket. As the rocket burns fuel, its mass decreases, causing a change in its velocity. This change in mass and velocity can be calculated using the rocket equation.

## 5. How is the variable mass problem different from the constant mass problem?

The variable mass problem involves objects whose mass changes over time, while the constant mass problem deals with objects whose mass remains the same. Additionally, the equations and principles used to solve these problems are different, as the variable mass problem requires the consideration of changing mass and its effect on an object's motion.