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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?

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 P. 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 P. Recall that the potential energy in a spring is 0.5*k*Δ², where Δ is the stretch/compression in the spring. Δ is NOT equal to the length of the spring. Recall that the spring is unstretched at position 2.

Step 3: Kinematics
What kinematics do you need here?

Step 4: Solve
Solve your work/energy equation for the speed of P at position 2.

Any questions??

Discussion and hints:

The constant applied force F acts only in the x-direction. Therefore, as explained in the lecture book, the work done by F is simply the force multiplied by the distance traveled by particle A in the x-direction. Use this result in the Step 2 of your analysis.

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+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. The KE for the system is the sum of that for A and B. The PE for the system is the sum of that for A and B. Consider the above discussion when calculating the work done by the force F.

Step 3: Kinematics
What is the horizontal distance traveled by B? How are the speeds of A and B related to the angular speed of the bar?

Step 4: Solve
Solve your work/energy equation for the angular speed of the bar.

Any questions??

Any questions??

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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).

Any questions??

As P moves around on the circular track, two things occur:

1. The normal force N on P due to the circular guide is proportional to the centripetal acceleration  of P: N = mv²/R.
2. A friction force opposes its motion, where the sliding friction force is proportional to the normal force between the circular guide and P: f = μ_kN = mμ_kv²/R.

From this, we see that the friction force goes to zero as the speed goes to zero. What does this imply about P coming to rest? Can you see this in the animation of the motion below?

HINTS:
You will need to use the chain rule of differentiation to set up this problem: dv/dt = (dv/ds)(ds/dt) = v (dv/ds).

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Discussion

FOUR-STEP PLAN

Step 1: FBD - Draw a free body diagram of particle P. Note that the slot is smooth (no friction), and that the system moves in a horizontal plane (no influence of gravity).

Step 2: Newton - Recommended that you use a set of xy-coordinate axes attached to slotted arm. Resolve your forces into xy-components, and write down Newton's 2nd law for P in terms of its xy-components.

Step 3: Kinematics - It is recommended that you use the moving reference frame velocity and acceleration equations for point P, with the observer being attached to the slotted arm:
v_P = v_O + (v_P/O)_rel + ω x r_P/O
 a_P = a_O + (a_P/O)_rel + α x r_P/O + 2ω x (v_P/O)_rel + ω x (ω x r_P/O)
Along with the rigid body velocity and acceleration equations for link AP, you will be able to solve these equations for the xy-components of the acceleration of P.

Step 4: Solve - Combine your equations from Steps 2 and 3 to solve for the normal force acting on P by the slot and the force on P by link AP.

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Discussion

FOUR-STEP PLAN

Step 1: FBD - Draw individual free body diagrams of A and B, along with an FBD of pulley C.

Step 2: Newton - From each FBD, write down the Newton's equation for components along the incline. Recall that the pulley has negligible mass.

Step 3: Kinematics - You will need to use the cable-pulley system kinematics that we worked with earlier in the semester. Please review the material from Section 1.D of the lecture book to relate the accelerations of blocks A and B.

Step 4: Solve - Combine your equations from Steps 2 and 3 to solve for the accelerations of blocks A and B.