# Homework H5.G - Fa22

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 - Fa22

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 work for 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 - Fa22

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 - Fa22

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 - Fa22

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 - Fa22

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.

# Homework H5.A - Fa22

Any questions??

Four-step plan:

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

Step 2 - Kinetics: Write down the Newton/Euler equations for the crate. Be careful with your moment equation. Since the crate has no fixed points, you need to sum moments about the center of mass of the crate.

Step 3 - Kinematics: Crate does not tip, so α = 0.

Step 4 - Solve

# Homework H5.B - Fa22

Any questions??

For Part (a), the coefficient of static friction between the sphere and the ramp is sufficiently large that the sphere does not slip as it rolls. Here f ≠ μkN.

For Part (b), the coefficient of friction is reduced such that slipping does occur. Here f = μkN.

Can you see the difference in rotational motion of the sphere between the no-slip and slipping cases?

Four-step plan:

Step 1 - FBD: Draw a free body diagram of the sphere. Take care in getting the direction of the friction force on the sphere correct.

Step 2 - Kinetics: Write down the Newton/Euler equations for the sphere. Be careful with your moment equation - it is recommended that you use a moment equation about the center of mass of the sphere.

Step 3 - Kinematics: For Part (a), C is a no-slip point. Relate the angular acceleration of the sphere to the acceleration of G. For part (b), C is NOT a no-slip point. You cannot relate the angular acceleration back to the acceleration of G through kinematics. Instead you need to use: f = μkN.

Step 4 - Solve

Any questions??

# Homework H4.T - Fa22

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: vP2 = vA2.
STEP 4 - Solve.