Any questions? Ask/answer questions in the discussion thread below.

DISCUSSION

The animation below is for the case of c = 0 (undamped).

The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of θ(t).

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

Step 1: FBDs
Draw individual free body diagrams for the drum and the block. Choose a translational coordinate (say x, defined as being positive to the left).  Be sure the get the correct direction for the spring and dashpot forces on the block. Also, take care in drawing the friction force on the drum as being equal and opposite to the friction force on the block.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the drum and block based on your FBDs above. Be sure to be in consistent in your sign conventions for forces/translation and moments/rotation.

Step 3: Kinematics
The contact point of the drum on the block (call it point A) is a no-slip point; that is, the horizontal component of acceleration of A is equal to the acceleration of the block.

Step 4: EOM
Combine your Newton/Euler equations along with your kinematics to arrive at a single differential equation in terms of the dependent variable θ.

We encourage you to interact with your colleagues here in conversations about this homework problem.

DISCUSSION
As always, we should follow the four-step plan:

STEP 1: FBD
Draw an FBD of the bar. Since the support at B is smooth, the reaction on the bar at B will be perpendicular to the bar.

STEP 2: Kinetics
You should write down the two Newton equations, and the Euler equation. As always, take care in choosing your reference point for Euler's equation. Since we have no fixed points for the bar, you should choose the center of mass G as your reference point.

STEP 3: Kinematics
Note that the path of the center of mass G is tangent to the surface of the bar (that is, the bar can move only along the support, not into the support). With the bar being released from rest, the acceleration of G is tangent to the path. Specifically, the acceleration of G is directed along the direction of the bar. Write down the rigid body kinematics equation that relates the accelerations of A and G. This vector equation represents two scalar equations (x- and y-components).

STEP 4: Solve
At this point, you will have three kinetics equations and two kinematics equations, for a total of five equations. You will have five unknowns. Solve!

Any questions?

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

DISCUSSION

Four-step plan

Step 1: FBD
Draw a free-body diagram particles A and B, along with the bar, altogether. Is angular momentum for this system conserved? Why, or why not?

Step 2: Kinetics - Angular impulse/momentum
Write down the angular momentum for A, B and the bar individually, and then add together. For a particle A, for example, recall that angular momentum is given by H_O = m r_A/O x v_A.  For a rigid body, H_O = I_Oω. You will also need the coefficient of restitution equation relating the radial (normal components here) components of velocity of A and B during impact.

Step 3: Kinematics
For writing down the velocity of points A and B, a polar description is recommended. For example, v_A = R_dot e_R + Rωe_θ.

Step 4: Solve
Solve your equations from Steps 2 and 3 for the velocity of P. Do not forget that A will have both radial and transverse components of velocity.

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

Discussion and hints

The four-step plan:

1. FBD: It is recommended that you draw a free body diagram of the plate and cart combined.
2. Kinetics: Your FBD should show that the summation of forces in the direction of motion for the cart (call that the x-direction) is zero; therefore, linear momentum in the x-direction is conserved for ALL time. Also note that energy is conserved up to the point of impact. During impact, energy is not conserved; however, you are given the coefficient of restitution (COR) for the impact.
3. Kinematics: Write down the rigid body velocity equation relating the motion of points O and the plate's center of mass G. Here you will use the fact that the cart moves only in the x-direction.
4. Solve

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

Discussion

.

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: v_A = ω x r_A/C, where C is the point on the disk in contact with the ground.
STEP 4: Solve

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

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

Discussion

The animation shown above demonstrates the kinematics of the motion of the bar/disk system. The bar rotates about end O. The disk is pinned to end A of the bar, and rolls without slipping on the fixed circular surface. Since the disk rolls without slipping, when point C on the outer circumference of the disk is in contact with fixed ground, the velocity of C is zero. You can readily see this in the animation above.

The instant center (IC) for the disk is the contact point of the disk with the fixed surface. Note the relative sizes of the speeds of A and C in comparison to their distances from the IC. When are the speeds the same? At what positions is the speed of C twice that of the speed of A?

You are asked to determine the rotation rate of the bar when the bar reaches the vertical position shown below.

HINT: As always, we should follow the four-step plan for solving this problem.
STEP 1: FBD. We will be using the work/energy equation to determine this angular speed. Based on earlier recommendations, we will make the choice of our system BIG, including the bar, and disk together.
STEP 2: Kinetics (here, work/energy). The total kinetic energy of the system shown in your FBD above is that of the bar + that of the disk. For the bar, it is recommended that you choose the fixed point O as your reference point for the KE. For the disk, it is recommended that you choose the center of mass A.  Be sure to identify the datum line for the gravitational potential energy, and use this in writing down this potential.
STEP 3: Kinematics. You need to relate the speed of A to the angular speeds of the bar and of the disk.
STEP 4: Solve

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

Discussion

The animation shown above demonstrates the kinematics of the motion of the system made up of bars OA and AB. Bat OA rotates about the fixed point O. Bar AB is attached of OA with a pin at A, and end B is constrained to move along in a smooth, horizontal slot.

You are asked to determine the speed of end B of AB when the bar OA reaches the vertical position shown below.

QUESTION: Where is the instant center (IC) of AB for the position above? What are the consequences of this location for IC_AB? Can you see these  in the above image when you consider the relative sizes of the speeds of points A, G and B?

HINT: As always, we should follow the four-step plan for solving this problem.
STEP 1: FBD. We will be using the work/energy equation to determine this angular speed. Based on earlier recommendations, we will make the choice of our system BIG, including both bars together.
STEP 2: Kinetics (here, work/energy). The total kinetic energy of the system shown in your FBD above is that of the bar OA + that of bar AB. For bar OA, it is recommended that you choose the fixed point O as your reference point for the KE. For bar AB, it is recommended that you choose the center of mass G for the reference point for its KE.  Be sure to identify the datum line for the gravitational potential energy, and use this in writing down this potential. What is the work done by force F?
STEP 3: Kinematics. Locate the IC for bar AB. What does this say about the angular speed of AB? Use this result in solving your energy equation.
STEP 4: Solve

Ask your questions here. Or, answer questions of others here. Either way, you can learn.

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 (FBD) of the bar and the particle.

Step 2: Kinetics (Work/energy)

• Looking at your FBD above, which forces, if any, do work that is not a part of the potential energy of the system? If there are none, then energy is conserved.
• Write down the individual contributions to the kinetic energy from the bar and particle, and add together for the total kinetic energy of the system.
• Define your gravitational datum line. Write down the individual contributions to potential energy from the bar, particle and spring, and add together for the total potential energy.

Step 3: Kinematics
Use the following rigid body kinematics equation to relate the angular velocity of the bar to the velocity of the particle B at position 2:

v_B= v_G + ω_AB x r_B/G

Step 4: Solve
From your equations in Steps 2 and 3, solve for the angular velocity of the bar.