How Do You Label Forces in Multi-Body Systems?

[Miike012]

I know how to create the force diagram but when the system is created of two or more bodies in motion I don't know how to properly create the formulas.

For instance look at the diagram I made of two bodies moving..

1. Ʃ (torque) = T1(R) - T2(R) = (I)(a/R) = (MR^2/2)(a/R) = M*a*R/2

2. Ʃ(F1) = T1 - w1 = (m1)(a)

3. Ʃ(F2) = w2 - T2 = (m2)(a)

With the upward and downward acceleration I am confused how to create my formulas...

 

[grzz]

the equation for the moment (torque) is correct.

Since the mass m2 is accelerating upwards one uses F = Ma in the upwards direction
i.e. T2 - W2 = m2a

and mass m1 is accelerating downwards one uses F = ma in the downwards direction
i.e. W1 - T1 = m1a.

 

[Miike012]

This was my thought process... imagining a free falling body accelerating downward, its weight vector points down... that's why I don't understand if body one is accelerating downward why is w1 positive??

 

[grzz]

re F = ma for body 1

we have W1 - T1 = m1a.

m1 is accelerating downwards and so we have to find the resultant (net) force F downwards.

i.e. we have to find which forces 'help' it to accelerate ...like W1 and so W1 is positive showing that W1 is 'helping' m1 to accelerate and which forces do not 'help' this acceleration ... like T1 and so T1 is negative.

 

[asteriatic]

Thank you for your question. Creating force diagrams and equations for systems of multiple bodies can be challenging, but with some practice and understanding of the principles involved, it can become easier.

First, it is important to understand that forces can be broken down into component forces, such as horizontal and vertical forces. This can help simplify the analysis of the system.

In your diagram, you have correctly labeled the forces and torques acting on the two bodies. To properly create the equations, you can use Newton's second law, which states that the sum of all forces acting on an object is equal to its mass multiplied by its acceleration (ΣF = m*a). This law can be applied to each body individually, and also to the system as a whole.

For example, in your second equation, you correctly wrote Σ(F1) = T1 - w1 = (m1)(a). This equation represents the sum of all forces acting on body 1. The first term, T1, represents the tension force acting on body 1, and the second term, w1, represents the weight force acting on body 1. These two forces are in opposite directions, so they are subtracted from each other. The resulting force is equal to the mass of body 1 multiplied by its acceleration.

Similarly, in your third equation, Σ(F2) = w2 - T2 = (m2)(a), you have correctly represented the forces acting on body 2. The weight force, w2, is acting downward, while the tension force, T2, is acting upward. Again, these two forces are subtracted from each other, and the resulting force is equal to the mass of body 2 multiplied by its acceleration.

To account for the upward and downward accelerations, you can use the principle of equilibrium, which states that for an object to be in equilibrium, the sum of all forces acting on it must be equal to zero. This can help you create equations for the system as a whole.

In your first equation, you have correctly written the equation for torque, Σ(torque) = T1(R) - T2(R) = (I)(a/R). This equation represents the net torque acting on the system, which is equal to the moment of inertia (I) multiplied by the angular acceleration (a/R). To account for the upward and downward accelerations, you can multiply the angular acceleration by the radius (R) to