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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?

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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. 

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?

Discussion and hints:

Let’s first take a look at the motion of the mechanism, as shown in the simulation results below.

The kinematics for this problem are a little complicated. Recall that we studied these types of “constrained motion” kinematics problems back in Section 1.D of the lecture book. Here, we will now get to use the skills developed there in solving this problem in kinetics.

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

Step 1: FBDs
Draw individual free body diagrams for bodies A and B.

Step 2: Kinetics (Newton’s 2nd Law)
Write down the appropriate Newton’s 2nd law equations for blocks A and B.

Step 3: Kinematics
Here, we will use the assumption that the cable length, L, does not change as the system moves (the cable does not stretch, go slack or break). For this, write L in terms of the motion variables s_A and s_B. You might want to review the material and examples from Section 1.D of the lecture book in doing this. Differentiate this relation once to relate the speeds of A and B, and then, once again, to relate the accelerations of A and B. NOTE: Be sure to use the same sign conventions for forces as for accelerations. For example, if you choose positive x as being to the right, then a_A = – s_A_ddot.

Step 4: Solve
At this point you will have three equations involving the tension in the cable, the acceleration of A and the acceleration of B. Solve these three equations for those three unknowns.

Any questions??

Discussion and hints:

Since all motion is in a horizontal plane and with the contact surface of the slot with P being smooth, the speed of P will remain constant throughout. And, with the speed being a constant, there will be only a normal (centripetal) component of acceleration for P. (If you do not see this from the beginning, no worries. Your analysis will show you this.) This is seen in the animation provided below.

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

Step 1: FBDs
Draw a free body diagram for particle P for the position of interest: x = 0. Please be reminded that the motion of the system is in a horizontal plane. With this and with the slot being smooth, only the normal contact force from the slot acts on P.

Step 2: Kinetics (Newton’s 2nd Law)
Write down the t – and n -components of the Newton’s 2nd law equations for P.

Step 3: Kinematics
Here, we will consider the constrained motion of particle P. Use the chain rule to determine y_dot in terms of x_dot, as well as y_ddot in terms of x_ddot. For the instant of interest, i = e_t and j = e_n.

Step 4: Solve
From your kinetics and kinematics equations, solve for the normal contact force, N.

Any questions??