Equations of Motion Using Newton's Method --

[Erikono]

Homework Statement

Determine the equations of motions in terms of x and gamma.
Assume small angles and that the wheel rolls without slip. The mass of the thin homogeneous large disk

of radius 2R is 2m. The mass of the thin homogeneous inner disk of radius R is m. The rod of length 2R is massless and rigid. The two pulleys are massless.

I have attached my homework set and the problem is number 15. (I wouldn't be able to describe this very well)

Homework Equations

We are supposed to use Newtons method.
See uploaded images for the answer and relevant equations.

The Attempt at a Solution

See other uploaded image for my attempt at the solution.

I'm confused as to what equations from what free body diagrams I should be using. Also, I don't know if the substitutions I am making or the constraints I have set up for the problem are right.

x_1 = R*theta ( displacements for wheels on top)
x_2 = x_1 - 2R*theta - 2R*gamma
F_s1 = k(x_1 + r*theta) (spring attached to wall)
F_s2 = T_1 = k(x_1 + r*theta) (spring between pulleys)
T_2 = mg
T_2x = T_2*gamma
F_friction = ?
I also summed moments about A which is where the smaller disk makes contact with the ground.

Any help on the matter would be greatly appreciated.

 

[anathema]

Thank you.

There are a few things to consider when solving this problem. First, you should draw free body diagrams for each individual element in the system (i.e. the two wheels, the rod, the two pulleys, and the two disks). This will help you identify all the forces acting on each element and how they are related to each other. From there, you can use Newton's second law (F=ma) to write equations of motion for each element in terms of x and gamma.

One important thing to note is that the two wheels are connected by the rod, so their motions are related. This means that the equations of motion for the two wheels will be similar, with some differences based on their individual masses and radii.

Additionally, you should consider the constraints given in the problem (i.e. small angles and no slip) when writing your equations of motion. These constraints will help you simplify the equations and make them easier to solve.

Once you have written the equations of motion for each element, you can use the equations to solve for the unknown variables (i.e. x and gamma). You may also need to use some of the relevant equations given in the problem (such as the relationship between displacement and angular displacement for a rolling wheel).

I hope this helps get you started on solving the problem. Good luck!