NOTE: A COR value of e = 0.8 was used in the simulation that produced the animated GIF above. The after-impact response that you predict with e = 0 will be different.

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HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBD: Draw free body diagram of the combined system of the plate + cart. Remember, make your system big.
2. Kinetics: What are the forces acting on the system in the horizontal direction? What does that say about the linear momentum of the system in that direction. Also, for Part a), energy is conserved (before impact). Conservation of linear momentum and conservation of energy will be two equations for you to use. The remaining equation(s) need to come from kinematics, as always.
3. Kinematics: What are the kinematics of the system immediately before impact?
4. Solve

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HINTS: Note the similarity of this problem to Homework 4.S that you worked earlier in the semester. Review that solution before starting this problem. As you review, take note of what you will need to do differently here than before. (The answer is: not much needs to be changed...) You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBD: Draw free body diagram of the combined system of the bar + particle. Remember, make your system big.
2. Kinetics: Consider using the moment about point O on the system during impact. What does this say about the angular momentum about point O? Also, before impact, energy is conserved. During impact, use the COR equation.
3. Kinematics: ??
4. Solve

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HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBD: Draw free body diagram of the disk.
2. Kinetics: Write down the summation of forces in the horizontal direction, and the summation of moments about the c.m. A. Note that during slipping, the friction force is mu_k*normal force. Use these in the impulse/momentum equations. This will give you two equations in terms of three unknowns (v_A2, omega2 and Delta_t).
3. Kinematics: Watch the animated GIF above of the motion of the disk. During the sliding portion, there is no kinematic relationship between the velocity of the disk center A and the angular speed of the disk. However, once slipping ceases, what is the relationship between the velocity of A and angular velocity of the disk? Think no-slip. (Be careful with the SIGN!) This is the third equation that you need for solving.
4. Solve

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

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HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBDs: Draw a free-body diagram of the total system (both bars together). Are there any non-conservative forces that do work on the system?
2. Kinetics: Write down the kinetic and potential energies of the system at the initial state, and the kinetic and potential energies of the system at the final state, along with any work done on the system by non-conservative forces. Using these, write down the work/energy equation.
3. Kinematics: At Position 2, link OA is vertical. The velocity vectors for A and B are parallel (watch the above video). What does that say about the IC of AB? And about the angular velocity of AB? And how the velocity of A is related to the velocity of the c.m. of AB?
4. Solve: With the above, you will have enough equations to solve for the unknowns.

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

Any questions??

HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBDs: Draw a free-body diagram of the total system (bar + disk). Are there any non-conservative forces that do work on the system?
2. Kinetics: Write down the kinetic and potential energies of the system at the initial state, and the kinetic and potential energies of the system at the final state, along with any work done on the system by non-conservative forces. Using these, write down the work/energy equation.
3. Kinematics: Use the rigid body velocity equation for the bar to relate the velocity of A to O. Similarly, use the rigid body velocity equation to relate the velocity of A to the no-slip contact point.
4. Solve: With the above, you should have enough equations to solve for the unknowns.

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

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CLARIFICATION: The distance d is measured along the incline. This is NOT the vertical change in height.

HINTS: The kinematics is the most challenging part of this problem, particularly the calculation of the distance though which the end of the cord moves (for finding work). To answer this, first consider that the contact point C is the INSTANT CENTER of the drum (no slip = zero velocity). Watch the above animation showing the velocity vectors for select points on the drum. Since C is the IC, then the velocity of point A on the drum (and, therefore, of the end of the cord) is THREE times the velocity of the drum center O. Question:  If the end of the cord moves three times as fast as O, what does this say about the distance traveled by the end of the cord compared to the distance traveled by O?

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

Any questions??

HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBDs: Draw a free-body diagram of the total system (bar + particle). Are there any non-conservative forces that do work on the system?
2. Kinetics: Write down the kinetic and potential energies at the initial state, and the kinetic and potential energies at the final state, along with any work done on the system by non-conservative forces. Using these, write down the work/energy equation.
3. Kinematics: You will need to be able to write the speed of the particle in terms of the angular speed of the bar. In addition, you need to find the stretched length of the spring at position 2.
4. Solve

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

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HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBDs: Draw individual freed-body diagrams of the blocks and the drum. Choose a set of coordinate axes. (For the discussion here, I will use x to the right, y up and z outward.)
2. Kinetics: For each FBD drawn, write down the Newton-Euler equations. This is a total of nine equations (three equations for each FBD). In actuality, you will likely only need THREE of these: summation of forces in y-direction for A, summation of forces in y-direction for B, and summation of moments for the drum. These three equations involve FIVE unknowns: the two tension forces, the accelerations of A and B, and the angular acceleration of the drum. REMINDER: Be sure that you follow the same sign conventions on forces as you do accelerations, and the same sign conventions on moments as you do angular accelerations.
3. Kinematics: Since you are short two equations, you need to use kinematics. Recommend that you use the rigid body acceleration equations to relate the accelerations of A and B to the angular acceleration of the drum. This gives you the two equations that you need to solve.
4. Solve: Solve the three equations for the three unknowns.

We are encouraging you to communicate with us and with your colleagues in the class through the threaded discussions on the course blog. If you have questions on this homework, please ask here.

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HINTS: You will find it helpful in solving this problem to follow the four-step plan outlined in the book:

1. FBDs: Individual FBDs for the disk and bar. When drawing the FBD of the bar, please note that the bar is NOT a two-force member with the reaction forces on the bar acting along the line connecting O and G; that is, the reaction at end O on the bar should be shown with Ox and Oy components. (Do you know why? If not, ask here on the blog for an explanation! Also, look at the "Two-force members" animation linked to from your lecture page.)
2. Kinetics: Write down Newton/Euler equations for each body. Be careful to use consistent sign conventions on forces/accelerations, and moments/angular accelerations. As you can see from the above animation, the angular acceleration of the bar is NOT the same as the angular acceleration of the disk.
3. Kinematics:  You will have too few equations for the number of unknowns from Step 2. Use the rigid body kinematics equations to relate the accelerations of O and G on the bar. Also, use the no-slip condition on the disk to relate the acceleration of O to the angular acceleration of the disk.
4. Solve