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DISCUSSION and HINTS

Initially Block A slides to the right along Block B which is traveling to the right. However, with friction acting between A and B, both A and B slow down. At some point, A instantaneously comes to rest, and the starts to move to the left. Once the speed of A to the left matches that of the speed of B to the left, the two stick and move together. You can see this in the animation that follows.

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

Step 1: FBDs
Draw single free body diagram (FBD) for the entire system (A+B). Do NOT consider A and B in separate FBDs because you will need to deal with the friction force acting between A and B (which you do not know).

Step 2: Kinetics (linear impulse/momentum)
Consider all of the external forces that you included in your FBD above. If there are no external forces acting in the horizontal direction (x-direction) on your system, the linear momentum in the x-direction is conserved.

Step 3: Kinematics
As described above, A comes to rest with respect to B when v_A = v_B.

Step 4: Solve
Combine your kinetics equation from Step 2 with your kinematics that you found in Step 3, and solve for the velocity of B.

QUESTION: Are you surprised that your answer for the final speed of B (and A) does not depend on the coefficient of friction acting between A and B? I was the first time that I worked the problem. 🙂

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DISCUSSION and HINTS
This problem is a straight-forward application of the linear impulse/momentum equation.

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

Step 1: FBDs
Draw single free body diagram (FBD) of the block

Step 2: Kinetics (linear impulse/momentum)
Calculate the impulse due to the external forces acting on the block. Set this equal to the change in linear momentum of the block.

Step 3: Kinematics
None needed here.

Step 4: Solve
Solve for the speed of the block at the specified time.

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DISCUSSION and HINTS

Note: The parameters used in the simulation that produced the above animation may be different from those appearing the problem statement this semester.

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

Step 1: FBDs
Draw single free body diagram (FBD) for the entire system. It is recommended that you include Block A, Block B, the spring and the cable.

Step 2: Kinetics (Work/energy equation)
Consider all of the external forces that you included in your FBD above. Which forces, if any, do non-conservative work on that system? If there are such forces, write down the work done by those forces. Note that the applied force P is one such force that does work. P acts at the free end of the cable, and since the force is constant, the work is simply P times the distance traveled by the end of the cable.

Write down the expressions for kinetic and potential energy for the initial and final states of the motion.

Step 3: Kinematics
The kinematics that you need at this step is the distance traveled by the end of the cable. As mentioned above, this distance is part of the work term done by P. Determining this distance requires a little trig.

Step 4: Solve
Combine your kinetics equation from Step 2 with your kinematics that you found in Step 3, and solve for the stiffness k of the spring.

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

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

Step 1: FBDs
Draw single free body diagram (FBD) for the entire system.

Step 2: Kinetics (Work/energy equation)
Consider all of the external forces that you included in your FBD above. Which forces, if any, do non-conservative work on that system? If there are no such forces, then energy is conserved. Write down the expressions for kinetic and potential energy for the initial and final states of the motion.

Step 3: Kinematics
At this step, you need to relate the speeds of blocks A and B at both the initial and final states. At the initial state, you should find a numerical value for the speed of B. For the final state, you need an expression that relates the two speeds. Use methods that we learned back in Section 1.D of the lecture book.

Step 4: Solve
Combine your kinetics equation from Step 2 with your kinematics that you found in Step 3, and solve for the speed of A.

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

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

Step 1: FBDs
Draw single free body diagram (FBD) for the entire system.

Step 2: Kinetics (Work/energy equation)
Consider all of the external forces that you included in your FBD above. Which forces, if any, do non-conservative work on that system? If there are no such forces, then energy is conserved. Write down the expressions for kinetic and potential energy for the initial and final states of the motion.

Step 3: Kinematics
At this step, you need to relate the speeds of blocks A and B at the second state. Consider using the instant center (IC) approach for the rigid link for position 2. Block B moves along a horizontal surface, and Block A rides on a vertical guide. Where do the perpendiculars to these two velocity vectors intersect? Take a look at the freeze-frame of the animation of the motion at state 2 shown below. Does that image agree with your IC analysis? And, what does this say about the speed of block B at that state?

Step 4: Solve
Combine your kinetics equation from Step 2 with your kinematics that you found in Step 3, and solve for the speed of A.

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

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).

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

Step 1 - FBD: Draw individual FBDs of blocks A and B.

Step 2 - Kinetics (Newton): Write down the Newton's 2nd Law equations for A and B from your FBDs above.

Step 3 - Kinematics: This part of the problem solution requires the most thought. Suppose that you have an observer that is moving along with block B. What is the direction of the motion of A that is seen by this observer. (HINT: The observer sees A moving in the direction along the inclined interface of the two blocks.) Using this, you can write:
a_A = a_B + a_A/B
or,
a_A*j = a_B*i + a_A/B*(cos(theta)*i + sin(theta)*j)
Use the two components of this vector equation to relate a_A and a_B.

Step 4 - Solve.

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

In order to draw the FBDs for the case with friction, you need to know first the direction of impending motion; that is, you need determine whether A moves UP the incline or DOWN the incline, since friction will oppose that motion. To determine the direction of impending motion, solve the problem for the friction-free case first (μ_k = 0). Once this direction is determined, rework the problem with friction.

Recall the following four-step plan outline in the lecture book and discussed in lecture. You will need to do this for both the friction-free case and the with-friction case.

Step 1: FBDs
Draw individual free body diagrams for the two blocks.

Step 2: Kinetics (Newton's 2nd Law)
Write down Newton's 2nd law for each block.

Step 3: Kinematics
Recall the work that we did in Chapter 1 in relating accelerations of two particles connected by a cable-pulley system.

Step 4: Solve
At this point you will have two equations in terms of two unknowns: the normal contact force and the tension force. Solve.

Ask and answer questions here. You learn both ways.

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

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

Step 1: FBDs
Draw a free body diagram (FBD) for the car. What is the difference between the FBDs for Parts a) and b)? They are not the same!

Step 2: Kinetics (Newton's 2nd Law)
Since the position of the car is known in terms of its path, a path description for resolving the forces acting on the car is recommended. What directions are e_t and e_n for this problem? You will need to use a third unit vector k for the vertical direction.

Step 3: Kinematics
Since you have used a path description for resolving forces, you should also use a path description for acceleration; that is, a_P = v_dot*e_t +(v²/ρ) e_n. 

Step 4: Solve
At this point you will have two scalar equations to solve.