Learning Objectives 1. Differentiate between contact forces and non-contact forces. 2. State and apply Newton's laws of motion. 3. Draw a free-body diagram of an object or multiple objects in

a system. 4. Construct net force equations in multiple dimensions for objects based on their accelerations and free-body diagrams. 5. Solve net force equations symbolically in terms of given variables. 6. Analyze systems of objects and pulleys of negligible mass using a force approach. 7. Apply acceleration constraints. 8. Assess the validity of symbolic solutions by explicitly checking physical units 9. Make sense of symbolic solutions using proportional reasoning and sensemaking techniques. Problem Statement A block of mass m rests on an incline with angle and has coefficients of friction µs and Mk with the surface of the incline. It is connected via a massless string over a massless and frictionless pulley to a hanging block of mass M. ● Massless string. m a. (5 points) Setting Up the Problem M Massless, frictionless pulley. Choose an appropriate coordinate system(s) and draw free body diagram(s) for each block. Make a table of known/given information and unknown/wanted information. (You may need to read the later parts of the problem to identify this) List the physical assumptions you are using to solve the problem. b. (5 points) Setting Up Equations: Write down Newton's Second Law equations for both block m and block M assuming that mass m is just large enough (or M is small enough) that the whole system remains at rest. c. (5 points) Solving your equations Use your system of equations to determine a symbolic expression for the minimum mass m that will stick to the surface and not slip. ● d. (5 points) Modifying the scenario: if the minimum mass is nudged ever so slightly to set it in motion up the incline, it will start moving up the incline. What is the acceleration constraint for the two blocks? ● e. (5 points) Sensemaking: Assess the validity of your solutions in each of the following ways: Unit Check: Check the physical units of your expression for the acceleration. Limiting Case 1: Consider the situation where m << M (mass M is many, many orders of magnitude greater than mass m... for example, think lead block versus Styrofoam block). O What do you expect the approximate acceleration of block M to be? O Does your symbolic solution from part d match your expectations in this situation? O Given your coordinate system does the sign of your acceleration make sense? ● ● Find a symbolic expression for the accelerations of block m and block M in this scenario. (Hint: Which expressions from part (b) valid are still valid in this new case? Which ones need refined?) ● Limiting Case 2: Consider the situation where m >> M. o What do you expect the approximate acceleration of block m to be? O Does your symbolic solution from part d match your expectations in this situation? O Given your coordinate system does the sign of your acceleration make sense?