Discussion and hints:

Let’s first take a look at the motion of the mechanism, as shown in the simulation results below.

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBD
Draw a free body diagram for the entire system of A+B+bar. Which, if any, forces do non-conservative work on this system? Can you justify this from the FBD?

Step 2: Kinetics (work/energy equation)
Write down the work energy equation for the system of A+B+bar. The KE will be the sum of the KEs of A and B. The PE will be the sum of the PEs for A and B.

Step 3: Kinematics
Shown below is a freeze frame of the system at position 2. Where do you see the IC for AB to be located? Use the location of the instant center of AB at Position 2 to simplify your kinematics for that position.

Step 4: Solve
Solve your work/energy equation for the speed of block A.

Any questions??

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HINTS

STEP 1 – FBD: Draw a SINGLE free body diagram (FBD) of the system including Block A, Block B and the cable. From this, determine which forces do work on this system.
STEP 2 – Kinetics:  Write down the work/energy equation. Determine the work done by the forces that you identified  above in STEP 1.
STEP 3 – Kinematics: Review the constrained motion kinematics from Section 1.D of the course lecture book. To this end, you will write down an expression for the length of the cable in terms of s_A, s_B and constants. Differentiate this expression to relate the speeds of A and B.
STEP 4 – Solve

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HINT: It is recommended that you review page 211 of the lecture book regarding the determination of the work due to an applied force using Cartesian coordinates. Using this can greatly simplify your analysis.

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DISCUSSION

Particle P moves in a way that it is constrained to move along the parabolic-shaped guide, as well as within the vertical slot. Your task here is to determine the reaction forces acting on P by the guide and the slot that are needed to enforce these motion constraints.

As you watch the animation above for the motion of P, why is the acceleration of P always pointing in the y-direction?

Hints:
You should follow the four-step solution plan described in the lecture book, and as discussed in lecture:
Step 1: Free body diagram (FBD) – Draw an FBD of P alone.
Step 2: Kinetics – Write down the Cartesian components of Newton’s 2nd law for P.
Step 3: Kinetics – You need kinematics here to related the known x-components of velocity and acceleration of P to its y-components.
Step 4: Solve – You will have two equations and two unknowns. Solve these for the two reaction forces.

Ask and answer questions below. You will learn from both asking and answering.

Any questions??

As P moves around on the circular track, two things occur:

1. The normal force N on P due to the circular guide is proportional to the centripetal acceleration  of P: N = mv²/R.
2. A friction force opposes its motion, where the sliding friction force is proportional to the normal force between the circular guide and P: f = μ_kN = mμ_kv²/R.

From this, we see that the friction force goes to zero as the speed goes to zero. What does this imply about P coming to rest? Can you see this in the animation of the motion below?

HINTS:
You will need to use the chain rule of differentiation to set up this problem: dv/dt = (dv/ds)(ds/dt) = v (dv/ds).

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Discussion

FOUR-STEP PLAN

Step 1: FBD – Draw individual free body diagrams of A and B, along with an FBD of pulley C.

Step 2: Newton – From each FBD, write down the Newton’s equation for components along the incline. Recall that the pulley has negligible mass.

Step 3: Kinematics – You will need to use the cable-pulley system kinematics that we worked with earlier in the semester. Please review the material from Section 1.D of the lecture book to relate the accelerations of blocks A and B.

Step 4: Solve – Combine your equations from Steps 2 and 3 to solve for the accelerations of blocks A and B.

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In the animation of the simulation shown below, the RED vectors shown are the forces of reaction acting on particles A and B (such as the force on each particle by member AB, and the normal forces of reaction by the floor and wall).

Since the motion of P is being described here in terms of polar variables of r and θ, it is recommended that you use a polar description for your forces and acceleration.

Use the Four-Step solution plan outlined in the lecture book:

Step 1 – FBD: Draw a free body diagram of C. NOTE: The arm rotates about a vertical axis, meaning that the arm moves in a horizontal plane; that is, the gravitational force acts perpendicular to the plane of the paper.

Step 2 – Kinetics (Newton): Resolve the forces in your FBD into their polar components. Sum forces in the r-direction and set equal m*a_r. Sum forces in the θ-direction and set equal to m*a_θ

Step 3 – Kinematics: Use the polar kinematics descriptions of a_r = r_ddot – r*θ_dot^2 and a_θ = r*θ_ddot + 2*r_dot*θ_dot.

Step 4 – Solve. When solving for the normal force, N, acting on C take note of the sign on your answer. What does this sign mean in terms of answering Part (c)?

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Please note that since P is not sliding on the rotating guide, P is traveling along a horizontal circular path having a radius with the radius r being the perpendicular distance from P to the vertical shaft. It is recommended that you use a set of polar coordinates: e_r pointing outward from the vertical shaft to O; e_φ tangent to the above-described path of P; and, k pointing upward.

Use the Four-Step solution plan outlined in the lecture book:

Step 1 – FBD: Draw a free body diagram of P. With the guide being smooth, there will be only two forces acting on P: the weight and the normal force N from the rotating guide.

Step 2 – Kinetics (Newton): Resolve the forces in your FBD into their polar components. Sum forces in the r-direction and set equal m*a_r. Sum forces in the k-direction and set equal to 0 (since P has no vertical motion for all time).

Step 3 – Kinematics: Use the polar kinematics descriptions of the acceleration of P. Note that r is constant for all time and Ω is constant.

Step 4 – Solve. With the above equations you will have sufficient number of equations to solve for the unknowns in the problem, which includes N and θ.

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