The presence of wheels on the cart means I want you to approximate this as + 20 frictionless. We will use a different coordinate system +Yc for each part of this physical setup. For the mc cart we are using the tilted coordinate * X E system labelled x-yc. For the hanging mass, my we will take downwards to be the positive Yu direction. For the pulley, I've labelled the +YH positive z direction as zo. I've chosen these coordinate systems such that, if the physical system moves, the sign of the motion will be the same for all parts of it-- that is, if the hanging mass moves in the +y, direction, then the cart will move in the +x, direction. It is possible to do the problem with different choices of coordinate systems, but somewhat harder, so this is advised. Considering theta, my, m., mc, and R, the radius of the pulley whose mass is m, to be known, our goal will be to find the (non-zero) acceleration of the hanging mass & cart (which is the same as each other since they are attached to each other with the rope and so move together (and both with the same sign because we strategically chose our positive directions to make it that way)). 1. Recall the moment of inertia of a disk around an axis through its center is (1/2) M R^2. Can the tension of the rope pulling down on the pulley have the same magnitude as the tension of the rope pulling parallel to the ramp? Why or why not? 2. Draw an extended force diagram for the pulley. Give each force in the diagram a sensible symbol distinct from all other symbols (and not a reserved letter like "g" which is only for 9.8N/kg near the surface of Earth, or "G" which is only for the universal gravitational constant). When in doubt, I suggest "F" with a subscript. Give a key that lists each symbol and tells what object is exerting the force as well as what object the force is acting on. 3. Draw a FBD for the hanging mass. Give each force a sensible symbol. (It's preferable for weight forces in different force diagrams to have different symbols, so subscript H for hanging mass, subscript c for cart, etc.) Give a key to the symbols as described previously. If any of the forces in this FBD can be related to forces in the extended force diagram for the pulley, state how and under what assumptions. (Notice that for objects that are not rotating, FBDs with just a point mass are sufficient, but if there is the potential for rotation, then an extended force diagram is needed because torques will have to be analyzed.)4. Draw a FED for the cart. Give each force a sensible symbol. Give a key to the symbols as described previously. If any of the forces in this FED can be related to forces in the extended force diagram for the pulley, state how and under what assumptions. 5. In symbols 81 vector notation, apply the momentum principle, Net Force = dpfdt, to each FED [for the hanging mass and for the cart), and apply the angular momentum principle, Net Torque = dedt, to the pulley. Looking at your three symbolic equations, list which symbols are unknowns. (Go back to the problem description to find what we said will be known.) 6. What symbolic equation expresses the relationship between the angular acceleration of the pulley and the linear acceleration of the hanging mass (which is the same as the acceleration of the cart}? 7. If you are up to the challenge, do symbolic algebra to find an expression for the acceleration of the hanging mass in terms of the knowns listed at the start of the problem. And whether you are up to the challenge just issued or not, you still need to find the numerical value of the acceleration of the cart for the case of rnH = 1 kg, rnp = 0.5 kg, m, = 1.5 kg, 8: theta = 20 degrees. If I did my work right, I didn't need the radius of the pulley. {If you are taking the symbolic algebra challenge, see if you agree with that.) Just in case there is a mistake in my work, let's take R to be 15 cm