Discussion and hints:

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

Step 1: FBDs
Draw a single free body diagram for the system made up of the disk and rod combined.

Step 2: Kinetics (work/energy)

• Write down the kinetic expression for the disk and rod individually, and then add those together to find the total KE for the system of your FBD. For each KE expression, recall that your reference point needs to be either the center of mass of the body, or a fixed point on the rigid body. You might consider using the no-slip, rolling contact point as your reference point for the disk.
• Do the same for the potential energies: write down the PEs for each body individually and add together.
• Also, based on your FBD above, which, if any force, does nonconservative work on the system in your FBD? Determine work for such a force.

Step 3: Kinematics
Note that the instant center (IC) for the disk is the no-slip contact point C. Where is the IC for rod OA at position 2? You might want to review Chapter 2 of the lecture book in finding the IC for a rigid body moving in a plane. (Carefully study either the animation above for position 2 or the freeze frame for that position below – you can actually see the IC from these!) Locating this IC is critical for you in setting up and using the kinematics for this problem.

Step 4: Solve
Solve your equations above for the velocity of point A.

Any questions?

Discussion and hints:

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

Step 1: FBDs
Draw a single free body diagram for the system made up of links OA and AB, along with slider B, combined.

Step 2: Kinetics (work/energy)

• Write down the kinetic expressions for the two links and the slider, individually, and then add those together to find the total KE for the system of your FBD. For each KE expression, recall that your reference point for the rotational component needs to be either the center of mass of the body, or a fixed point on the rigid body.
• Do the same for the potential energies, if needed: write down the PEs for each body individually and add together. Also, include the PE of the spring.
• Also, based on your FBD above, which, if any force or couple, does nonconservative work on the system in your FBD? Determine work for such a force/couple.

Step 3: Kinematics
Where is the instant center (IC) for link AB for position 2? (Please refer back to Chapter 2 of the lecture book if you need to review locating ICs.) What does the location of this IC say about the angular speed of link AB? And, what does it say about the speeds of points A and B? Does the freeze frame image of the mechanism shown below agree with your analysis?

Step 4: Solve
Solve your equations above for the speed of slider B.

Any questions?

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

This is a standard problem for the central impact of two bodies.

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

Step 1: FBDs
Draw three free body diagrams (FBDs): one of A alonw, one of B alone and one of A+B. Identify the n- and t-directions on your FBDs.

Step 2: Kinetics (linear impulse/momentum)

• In which directions, if any, is linear momentum conserved for A alone? For those direction(s), write down the appropriate momentum conservation equation.
• In which directions, if any, is linear momentum conserved for B alone? For those direction(s), write down the appropriate momentum conservation equation.
• In which directions, if any, is linear momentum conserved for A+B? For those direction(s), write down the appropriate momentum conservation equation.
• Recall that you also have the coefficient of restitution (COR) equation at your disposal. Keep in mind that the COR equation is valid for only the n-components of velocity.

Step 3: Kinematics
None needed here.

Step 4: Solve
From your equations solve for the n- and t-components of velocity of A and B.

NOTE: Please use F = 2000 N in this problem. 

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?

NOTE: Please ignore the numerical values provided for M and d – this is unintended extraneous information.

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?

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

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. 

Discussion and hints:

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

Step 1: FBDs
Draw two free body diagrams: one for each body individually.

Step 2: Kinetics (linear impulse/momentum)
Write down the linear impulse/momentum equation for each body individually. Note that the change in momentum of each particle is the same, and is equal to the area under the F(t) vs. t curve.

Step 3: Kinematics
None needed here.

Step 4: Solve
Solve for the speed of each body 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. 🙂

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?

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+P.

Step 2: Kinetics (linear impulse/momentum and work/energy)
From your FBD above, what is the external force acting on the system of A+P 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+P? What does this say about the mechanical energy of the system?

Step 3: Kinematics
The motion of P as seen by an observer on A is directed along the angle θ of the wedge. Write down the kinematics vector equation of v_P = v_A + v_P/A. This provides two scalar equations that represent the constraint of motion between A and P.

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

Question: Watch the animation above. Note that the actual path of P is at an angle that is different from the wedge angle θ – why is that?

Any questions?