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?

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?

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

DISCUSSION and HINTS

The animation below shows the impact of the particle with the rigid bar. As stated in the problem statement, the particle sticks to the bar during the short impact time.

Considering the system made up of the particle and the bar, we see that there are no fixed points that are easily recognized and determining the location of the center of the mass requires some calculation. Because of this, it is advisable to consider the particle and the bar in separate FBDs in your analysis.

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

Step 1: FBDs
Draw individual free body diagrams (FBDs) of the particle and bar. Be sure to draw the impact force on both FBDs.

Step 2: Kinetics (impulse/momentum)
Consider the linear impulse/momentum equation for the particle and the angular and linear impulse/momentum equations for the bar. Note that each of these equations will include the impulse of the impact force.

Eliminate the impulse of the impact force from the above three equations. This will leave you with two equations in terms of the post-impact velocity of the particle, the post-impact velocity of the bar’s center of mass, and the post-impact angular velocity of the bar.

Step 3: Kinematics
Since the particle sticks to the bar during impact, you can relate the post-impact velocities above through the rigid body kinematics equation:

v_B= v_G + ω_bar x r_B/G 

where B is the top point on the bar where the particle impacts and sticks.

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

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

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 bar.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the bar using your FBD above. Take care in choosing the reference point for your moment equation. In order to use the “short form” of Euler’s equation, this point should be either a fixed point or the body’s center of mass. For this problem, there are no fixed points.

Step 3: Kinematics
The paths of A and B are known: A travels on a straight path aligned with the inclined wall, and B travels on a circular path centered at O. Since the bar is released from rest, you know that the speeds of A and B are zero – therefore, the centripetal component of acceleration for each point is zero. This leaves the acceleration of points A and B tangent to their paths. (You can see this from the animation above for the instant when AB is horizontal.) It is recommended that you use two kinematics equations: one relating points A and B, and the other relating the center of mass G of the bar to either A or B.

Step 4: Solve
Solve your equations above for the tension in cable BO.

Any questions?

Discussion and hints:

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

Step 1: FBDs
Draw individual free body diagrams for bar OA and block B.

Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the block and the bar using your FBDs above. Take care in choosing the reference point for your moment equation for the bar. In order to use the “short form” of Euler’s equation, this point should be either a fixed point or the body’s center of mass. For this problem, you have a choice between using the center of mass of the bar or the pin joint O. Be reminded that if you use point O, you will need to use the parallel axis theorem when calculating the mass moment of inertia of the bar.

Step 3: Kinematics
Put some thought into the kinematics for this problem. This is the most complicated part of the solution. Importantly, the acceleration of A is NOT equal to the acceleration of B. Why? But then, how do you find the relationship between these two accelerations?

Step 4: Solve
Solve your equations above for the acceleration of block B.

Any questions?

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 and rod combined.

Step 2: Kinetics (work/energy)

• Write down the kinetic expression for the disk and rod 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 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 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. Where is the IC for rod OA at position 2? You might want to review Chapter 2 of the lecture book in finding the IC for a rigid body moving in a plane. (Carefully study either the animation above for position 2 or the freeze frame for that position below – you can actually see the IC from these!) Locating this IC is critical for you in setting up and using the kinematics for this problem.

Step 4: Solve
Solve your equations above for the velocity of point A.

Any questions?

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 links OA and AB, along with slider B, combined.

Step 2: Kinetics (work/energy)

• Write down the kinetic expressions for the two links and the slider, 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.
• Do the same for the potential energies, if needed: write down the PEs for each body individually and add together. Also, include the PE of the spring.
• 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
Where is the instant center (IC) for link AB for position 2? (Please refer back to Chapter 2 of the lecture book if you need to review locating ICs.) What does the location of this IC say about the angular speed of link AB? And, what does it say about the speeds of points A and B? Does the freeze frame image of the mechanism shown below agree with your analysis?

Step 4: Solve
Solve your equations above for the speed of slider B.

Any questions?

Ask and answer questions here. You learn both ways.

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?