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Homework H5.K - Fa24

Problem statement
Solution video

DISCUSSION THREAD

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


Discussion

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The animation shown above demonstrates the relationship between the translation velocity of the disk's center A and the angular velocity of the disk. Recall that the disk is initially projected to the right with an initial velocity of A and a counterclockwise rotation rate; that is, the disk is given a "backspin" on release.

As the disk slides, the friction with the floor does two things: it slows down the translational motion of the disk, and exerts a clockwise moment on the disk reducing the rotational rate of the disk. At some point the rotation rate changes from counterclockwise (positive) to clockwise (negative). And shortly after that, slipping ceases to exist between the disk and the floor. Rolling without slipping starts at the point where the plots of velocity of A and angular velocity of the disk go "flat".

 

HINT: As always, we should follow the four-step plan for solving this problem.
STEP 1: FBD. Here, draw a free body diagram of the disk.
STEP 2: Kinetics (here, linear and angular impulse momentum equations).
STEP 3: Kinematics. Slipping ceases when the velocity of the contact point on the disk with the ground goes to zero. At that point: vA = ω x rA/C, where C is the point on the disk in contact with the ground.
STEP 4: Solve


 

Homework H5.L - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Discussion and hints:

Note that the animation above is for a generic set of parameter values that may or may not be the ones assigned for the problem this semester.

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 bar OA and particle P, combined.

Step 2: Kinetics (impulse/momentum)

  • From you FBD above, what can you say about the moments acting about the fixed point O in your system? What consequence does this have on the angular momentum about point O for the system?
  • Use the coefficient of restitution, COR, equation relating the "n-components" of velocity for P and end A of the bar. (You may want to review Section 4.C of the lecture book in regard to impacts and the COR.)

Step 3: Kinematics
Note that bar OA rotates about point O. What does that say about the direction of the velocity of point A on the bar after impact? Study the animation above, and the freeze-frame images below of the motion. Do your kinematics agree with what you see about the direction of the velocity of A?

Step 4: Solve
Solve your equations above for the angular speed of bar OA.


Any questions?

Homework H5.I - Fa24

Problem statement
Solution video

DISCUSSION THREAD

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, bar and block, combined.

Step 2: Kinetics (work/energy)

  • Write down the kinetic expressions for the three bodies in the system, individually, and then add those together to find the total KE for the system of your FBD.
  • 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 or couple, does nonconservative work on the system in your FBD? Determine work for such a force/couple.

Step 3: Kinematics
Since the bar is welded to the disk, the angular velocities of the bar and disk are the same. The no-slip condition at the point of contact of the disk and block enforces a constraint between the rotation rate of the disk and the translational speed of the block. What is that constraint?

Step 4: Solve
Solve your equations above for the angular velocity of the disk.


Any questions?

Homework H5.J - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


DISCUSSION

Four-step plan

Step 1: FBD
Draw a free-body diagram for the plate.

Step 2: Kinetics - Work/energy
Write down the kinetic energy and potential energy expressions for the plate, along with the work done on the disk by non-conservative forces. The instant center, IC, is fairly easy to locate for Position 2 - you are able to write down the KE for the plate as Tplate = 0.5*IICω2, if you choose to do so. The work done by F is equal to U = FsD, where sD is the distance traveled by corner D of the plate.

Step 3: Kinematics
The easiest approach here is to use the location of the IC for the plate.

Step 4: Solve
Solve your equations from Steps 2 and 3 for the angular speed of the plate.

 

Homework H5.G - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Four-step plan

Step 1: FBD
Draw a free-body diagram for the disk.

Step 2: Kinetics - Work/energy
Write down the kinetic energy and potential energy expressions for the disk, along with the work done on the disk by non-conservative forces. Note that point C is the instant center (zero velocity point) for the disk. Because of this, you are able to write down the KE for the disk as Tdisk = 0.5*ICω2, if you choose to do so. The work done by F is equal to U = FsA, where sA is the distance traveled by end A of the cable.

Step 3: Kinematics
Here you need to relate the distance traveled by point O to the distance traveled by end A of the cable (these are not the same). In doing this, recall that C is the instant center of the disk as it rolls without slipping.

Step 4: Solve
Solve your equations from Steps 2 and 3 for the speed of point O.

 

Homework H5.H - Fa24

Problem statement
Solution video

DISCUSSION THREAD

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 block A, the disk and the pulley combined.

Step 2: Kinetics (work/energy)

  • Write down the kinetic expressions for the block, disk and pulley 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. You might consider using the no-slip, rolling contact point C 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 the work done by such a force.

Step 3: Kinematics
Note that the instant center (IC) for the disk is the no-slip contact point C. (Carefully study either the animation above - you can actually see the IC from this!) Locating this IC is critical for you in setting up and using the kinematics for this problem. What is the speed of point B on the disk as compared to the disk's center E? (Refer back to C being the IC for the disk.) See the freeze-frame image below, and compare the speed of B with that of A.

 

Step 4: Solve
Solve your equations above for the speed of block A.


Any questions?

Homework H5.E - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Four-step plan:

Step 1 - FBD: Draw a free body diagram of the bar.

Step 2 - Kinetics: Write down the Newton/Euler equations for the sphere. Since there is no fixed point on the bar, choose the center of mass for your moment equation

Step 3 - Use the rigid body kinematics equation for relating the accelerations of points A and B on the bar:

aB = aA + α x rB/A - ω2 rB/A 

followed by relating the acceleration of B to the acceleration of the bar's center of mass, G:

aG = aB + α x rG/B - ω2 rG/B 

Step 4 - Solve

 

Homework H5.F - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


DISCUSSION

Four-step plan

Step 1: FBD
Draw individual free-body diagrams for the plate and cart.

Step 2: Kinetics - Newton/Euler
Write down the Newton/Euler equations for the plate and the cart. Note that there are no fixed points observed for the plate; therefore, you need to use a moment equation of the plate about the plate's center of mass.

Step 3: Kinematics
Use the rigid body kinematics equations to relate the acceleration of the cart to the acceleration of the plate's center of mass through the angular acceleration of the plate.

Step 4: Solve
Solve your equations from Steps 3 and 4 for the angular acceleration of the plate and the acceleration of the cart.

Homework H5.C - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions??


Four-step plan:

Step 1 - FBD: Draw individual FBDs of the bar and the block.

Step 2 - Kinetics: Using your FBDs above, write down the Newton/Euler equations for the block and the bar. Be careful with your moment equation for the bar - since the bar does not have a fixed point, you need to use the center of mass of the bar for your moment equation.

Step 3 - Kinematics: You need to relate the acceleration of the block to the acceleration of the center of mass, G, of the bar through the rigid body acceleration equation:
aG = aO + αbar x rG/O - ωbar2 rG/O 

Step 4 - Solve

Homework H5.D - Fa24

Problem statement
Solution video

DISCUSSION THREAD

Any questions?? Please ask and answer questions in the threaded discussion below.


DISCUSSION: Four-step plan

Step 1: FBD
Draw individual free-body diagrams for the drum and block.

Step 2: Kinetics - Newton/Euler
Write down the Newton/Euler equations for the drum. Note that since the drum does not slip on either cable, point C on the drum is the instant center for the drum, with the acceleration of C, therefore, pointing toward the center of mass O. Because of this, you are able to use C for your Euler (moment) equation. This will simplify your analysis. Please note that if you do use C, you will need to use the parallel axis theorem in finding the mass moment of inertia of the drum about C.

Step 3: Kinematics
Use the fact that C is the center of rotation  in relating the kinematics of the drum to the kinematics of block B. Be careful in abiding by your sign conventions in this step. That is, if you chose the CCW direction to be positive for moments back in Step 2, this becomes the positive sign conventions for the angular acceleration. Similarly, if you chose the upward direction of B to be positive in the Newton equation for B, then that is the positive sign convention for the acceleration of block B.

Step 4: Solve
Solve your equations from Steps 3 and 4 for the angular acceleration of the drum and the acceleration of block B.