Simple Pulley System: Accelerations Equation Explained

[asi123]

Homework Statement

Hey guys.
I have the following pulley system (in the pic).
I wrote down all the equations.
The thing I don't get is the accelerations equation (a1 = 2*a2). I understand that I can get to this equation somehow by the length of the rope, but how?
Thanks.

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

The thing I don't get is the accelerations equation (a1 = 2*a2). I understand that I can get to this equation somehow by the length of the rope, but how?

Hi asi123!

You get it by double-differentiating x₁ = 2x₂ …

and that is simple geometry … when m₂ moves a distance x₂, m₁ moves a distance x₁ = 2x₂.

 

[thomate1]

I am happy to see that you are actively exploring and trying to understand the equations for a simple pulley system. To explain the accelerations equation, we need to first understand the principles behind a simple pulley system.

A simple pulley system consists of a rope or cable wrapped around a wheel or pulley, with one end attached to a fixed point and the other end attached to a load. When a force is applied to one end of the rope, it causes the load to move in the opposite direction. The motion of the load is dependent on the tension and direction of the forces acting on the rope.

Now, to understand the accelerations equation (a1 = 2*a2), we need to consider the motion of the pulley itself. In this case, the pulley is assumed to be massless, which means it has no mass and therefore no inertia. This allows us to simplify the equations and focus on the motion of the load.

Using Newton's Second Law, we know that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this case, the load is connected to two sections of rope, with one end attached to the fixed point and the other end attached to the pulley. This creates two forces acting on the load - the weight of the load pulling downwards and the tension of the rope pulling upwards.

Since the pulley has no mass, there is no net force acting on it and therefore it remains stationary. This means that the tension in the two sections of rope must be equal, as they are connected to the same stationary object. This also means that the acceleration of the load (a1) is equal to twice the acceleration of the rope (a2), as the load is being pulled by two sections of rope.

In summary, the accelerations equation (a1 = 2*a2) can be understood by considering the motion of the pulley and the forces acting on the load and the rope. I hope this explanation helps you understand the concept better. Keep up the good work in exploring and understanding scientific principles!