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Discussion and hints

The four-step solution procedure:

1. Step 1 – FBDs: Draw individual free body diagrams (FBDs) for A and B, along with an FBD of A+B. Since you have two impacts occurring at the same time, identify the “n” direction for each impact (they are different for each impact).
2. Step 2 – LIM and COR
   • For the impact of A with the bumper, use the coefficient of restitution (COR) equation.
   • For the impact of A with B, use conservation of linear momentum in the n-direction and the COR equation in the n-direction.
3. Step 3 – Kinematics: None needed.
4. Step 4 – Solve

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Animation

HINTS:

STEP 1 – FBD: Draw three free body diagrams (FBDs): A, B and A+B.
STEP 2 – Kinetics:  From the FBD of A, we know that it will continue to move along the same line of travel. In addition, the FBD of B shows that the t-component of velocity for B will remain unchanged, where the “t“-direction is tangent to the plane of contact of A and B. In the n-direction (perpendicular to the t-direction), you have both the linear-impulse equation and the coefficient of restitution equation.
STEP 3 – Kinematics
STEP 4 – Solve: Solve for M and e from your two kinetics equations above.

COMMENT: This problem is posed in a somewhat “backward” way as compared to many problems in impact. Typically mass and COR parameters are given, and we solve for final velocities. Here we are given some information on the final velocities, and we are asked to find mass and COR values. You use the same equations for either type of problem, you just solve these equations in a different way.

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HINTS:

STEP 1 – FBD: Draw a SINGLE free body diagram (FBD) of the system of cart + cannon + cannonball.
STEP 2 – Kinetics:  Write down the impulse/momentum equation in the horizontal direction (x-direction) for the the system of cart + cannon + cannonball. Based on the above FBD, is the momentum conserved in the x-direction for that system?
STEP 3 – Kinematics
STEP 4 – Solve. Solve for the velocity of the cart + cannon.

QUESTION: The above analysis allows you to find the answer to the first part of the problem. Unfortunately, it is not useful for the find the answer to the second part of the problem where you want to find the force on the cart/cannon. What do you need to change in the analysis to find this force?

Discussion and hints:

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

Step 1: FBD
Draw a free body diagram of the system made up of A+B.

Step 2: Kinetics (linear impulse/momentum and work/energy)
From your FBD above, what is the external force acting on the system of A+B in the horizontal direction? What does this say about the linear momentum of this system in that direction? Also, are there any non-conservative forces acting on the system of A+B? What does this say about the mechanical energy of the system?

Step 3: Kinematics
At position 2, B is moving only in the horizontal direction. There is no vertical component of velocity of B at position 2.

Step 4: Solve
Solve for the speeds of A and B from the above equations.

Any questions?

Ask and answer questions here. You learn both ways.

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:

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

__________________________

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