# Tension on the rope (classical mechanics problem)

• allshaks
In summary, the problem involves two mass points connected by a string passing through a hole in a smooth table, with one mass resting on the table and the other hanging suspended. The question is to find the tension in the string. Using the Euler-Lagrange equations and the constraint equation, we can set up equations involving the forces and accelerations of the masses. However, it is possible to solve the problem without using Lagrangian mechanics by simply drawing free body diagrams and analyzing the forces on the masses.
allshaks

## Homework Statement

The situation is that of Goldstein's problem 1.21 (or 1.19 in some editions):
"Two mass points of mass m1 and m2 are connected by a string passing through
a hole in a smooth table so that m1 rests on the table and m2 hangs suspended.
Assume m2 moves only in a vertical line."
But the question is the following: what is the tension?

## Homework Equations

The Euler-Lagrange equations of the system are:
$$\frac{d}{dt}(m_1 d^2 \dot{\theta})=0$$
$$(m_1+m_2)\ddot{\theta}=-m_2 g+m_1 d \dot{\theta}^2$$
The constraint equation is:
$$r_1-y_2=l$$
Where ##l## is the length of the rope.

## The Attempt at a Solution

I started by using the fact that the only force applied on ##m_1## is the tension, and that this tension must be in the direction of the string; that is, always radial (using the hole as the frame of reference). This means:
$$F_{r_1}=\frac{a_{r_1}}{m_1}= \frac{\ddot{r_1}-r_1 \dot{\theta}^2}{m_1}=-T$$
Where ##T## is negative because I define it as being positive when it is pulling up mass 2 (the other mass) such that:
$$T=m_2 \ddot{r_1}+m_2 g$$
Because ##\ddot{y_2}=\ddot{y_1}##
I've been playing with all of these equations for a while, but I couldn't find the solution. I'm not sure if there is a way of obtaining an expresion ##T## that involves Lagrangian mechanics, or if there are any other techniques to do it.
I'm following a problem set from a Classical Mechanics course I'm taking at college, and there is another question in the problem set: "In order to calculate the constraint forces on a system, what are the methods that could be employed?". I'm not sure how to answer that question, and I think that maybe some info on that (maybe some textbook that covers the subject?) may help me solve the problem. Thank you very much.

There is very little maths needed to solve this problem .

Draw free body diagrams for the two masses and just look at them .

## 1. What is tension on a rope?

Tension on a rope refers to the amount of force that is being applied to the rope in order to keep it in a state of equilibrium. It is a crucial concept in classical mechanics and is often used to analyze the forces acting on objects in motion.

## 2. How is tension calculated?

The tension on a rope can be calculated using the formula T = mg + ma, where T is the tension, m is the mass of the object, g is the acceleration due to gravity, and a is the acceleration of the object. This formula takes into account both the weight of the object and any additional forces acting on it.

## 3. What factors affect tension on a rope?

The tension on a rope is affected by various factors including the mass of the object, the acceleration of the object, the angle at which the rope is pulled, and any external forces acting on the object. These factors can change the magnitude and direction of the tension.

## 4. Can tension on a rope be greater than the weight of the object?

Yes, tension on a rope can be greater than the weight of the object. This can happen when there are external forces acting on the object, such as when an object is being pulled or pushed by another force. In this case, the tension on the rope will be equal to the weight of the object plus the additional force.

## 5. How does tension affect the motion of an object?

The tension on a rope can affect the motion of an object by either accelerating or decelerating it depending on the direction and magnitude of the tension. It can also change the direction of the object's motion if the tension is not in the same direction as the object's initial velocity. In some cases, tension can also keep an object in a state of equilibrium, preventing it from moving.

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