DISCUSSION

Since the problem asks for a relationship between the change of speed of P and the distance traveled by P, the work/energy equation is a natural method of choice. Recall that with the work/energy equation, we typically want to include as much in the system in order so that we can make as many forces to be workless, internal forces within the system. To this end, consider a system made up of P and the cable connected to P.

Hints:
You should follow the four-step solution plan described in the lecture book:
Step 1: Free body diagram (FBD) – Draw an FBD of the system made up of P and the cable. Which forces in your FBD do work on the system?
Step 2: Kinetics – Write down the work/energy equation for P, and the terms included in this equation.
Step 3: Kinetics – In order to determine the  work done on the system by the applied force F, you need to find the distance traveled by the end of the cable as P moves from position 1 to position 2. HINT: This distance is equal to the amount of cable that is pulled over the pulley as P moves from position 1 to position 2.
Step 4: Solve – Solve for the final speed of P at position 2.

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

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 of the system made up of B, bar AB and the spring.

Step 2: Kinetics (angular impulse/momentum and work/energy)
Note that all forces on the system act at the fixed point O. What does this say about the angular momentum of the system about point O? Also, consider the work/energy equation for the system.

Step 3: Kinematics
The kinematics of P are best written in terms of polar coordinates R and φ.

Step 4: Solve
Solve for the R and φ components of velocity of P from these equations.

Any questions?

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Discussion

You are asked to investigate the dynamics of this system during the short time of impact of P with A.

• It is suggested that you consider a system made up of A+P+bar (make the system “big”).
• Draw a free body diagram (FBD) of this system.
• For this system, linear momentum is NOT conserved since there are non-zero reaction forces at O.
• Furthermore, energy is NOT conserved since there is an impact of P with A during that time.
• From your FBD of the system, you see that the moment about the fixed point O is zero. What does this say about the angular momentum of the system about O during impact? (Answer: It is conserved!)

HINTS:

STEP 1 – FBD: Draw a SINGLE free body diagram (FBD) of the system of A+P+bar.
STEP 2 – Kinetics:  Consider the discussion above in regard to conservation of angular momentum about point O. Recall how to calculate the angular momentum about a point for a particle.
STEP 3 – Kinematics: At Instant 2, the P sticks to A: v_P2 = v_A2.
STEP 4 – Solve. 

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

Recall the definition of angular momentum of a particle P (of mass m) about a fixed point O:  H_O = m r_P/O x v_P.

For this problem, use this equation to find the angular momentum for each particle and add these together. As you work the problem, consider the number of cancellations that occur among these terms and consider why these cancellations occur. This will help you get insights on the meanings of angular momentum.

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 of P.

Step 2: Kinetics (angular impulse/momentum and work/energy)
Note that all forces acting on P in the plane of the table point toward  the fixed point O. What does this say about the angular momentum of P about point O? Also, consider the work/energy equation for P.

Step 3: Kinematics
The kinematics of P are best written in terms of polar coordinates R and φ.

Step 4: Solve
Solve for the R and φ components of velocity of P from these equations.

Any questions?

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

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

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

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.

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

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. 🙂