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DISCUSSION and HINTS

The animation below shows the impact of particle B with the bar.

Animation at slowed playback speed:

Free-frame of motion immediately after impact.

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 particle and bar.

Step 2: Kinetics (Newton/Euler)

Using your FBD above, sum moments about point A . Consider the time of the impact to be short such that there is no change in position of either the bar or B during impact. Also, consider the particle to be of small physical dimensions.

What does your moment equation above say about the angular momentum of the system about point A?

Step 3: Kinematics
What kinematics do you need to solve this problem?

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

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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 (FBD) of the disk.

Step 2: Kinetics (Newton/Euler)
Based on your FBD above, write down the linear impulse momentum and angular impulse/momentum equations for the disk.

Step 3: Kinematics
What kinematics do you need here to solve?

Step 4: Solve
From your equations in Steps 2 and 3, solve for angular velocity of the disk at time 2 and the reactions at O.

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DISCUSSION and HINTS

The animation below shows the motion of the disk as it moves down the incline. Included in the video are the friction and normal forces (FF and FN) acting on the disk as it moves. The direction of the friction force, as expected, opposes the direction of impending slip between the ramp and the disk.

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

Step 1: FBDs
Draw a free body diagram (FBD) of the disk. In drawing your FBD, please note that the friction force is NOT proportional to the normal force N; that is, f ≠ μN. Why is that?

It is recommended that you choose a set of coordinates that are aligned with the ramp. For example, choose the x-direction down the incline and the y-direction   perpendicular to the ramp pointing up and to the right.

Step 2: Kinetics (Newton/Euler)

• Based on your FBD above, write down the impulse/momentum equation in the x-direction for the disk.
• Based on your FBD above, write down the angular impulse/momentum equation for the disk.
• Combine the two equations above by eliminating the impulse of the friction force from the equations.

The above gives you a single equation in terms of two variables: v_O and ω_disk.

Step 3: Kinematics
Note that the no-slip contact point of the disk with the incline is the instant center (IC) of the disk. Let's call that point C. Since C is the IC of the disk, you can readily relate the angular velocity of the disk to the velocity vector of the disk center O through:

v_O= v_C + ω_disk x r_O/C = ω_disk x r_O/C

Be careful with signs.

Step 4: Solve
From your equations in Steps 2 and 3, solve for the velocity of point O on the wheel at time 2.

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Discussion

The animation shown below is a "flipped" version of the motion of our system here; that is, the vertical slot A below moves to the right, whereas in your problem it moves to the left. Otherwise, the motion is similar to your problem.

• For the range of motion shown below, P moves with positive x- and y-components of velocity. Can you see this in the animation?
• For all time, P moves with a positive x-component and a negative y-component of acceleration. Why? Can you see this in the animation?
• As we will see in the next class period when we talk about the "path description" of kinematics, the velocity of a point is always tangent to the path of the point, and the acceleration always points "inward" on the path of the point. Do you see this in the animation below?

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HINTS
The y-component of the position of P is not given to you explicitly as a function of time. Because of this, you will need to use the chain rule of differentiation in order to find the time derivative of y that is needed for the velocity of P:  dy/dt = (dy/dx)*(dx/dt). In this problem, as in others, do not confuse dy/dt with dy/dx.

DISCUSSION
Shown below is an animation created from the simulation of the motion prescribed for this problem. Some observations on this motion:

• The x-component of the velocity of P is a constant, as specified in the problem. Do you see this in the motion?
• The sign of the y-component of velocity of P changes from positive to negative over the course of the motion. Can you see this in the motion? Do you see this in your analysis? At what value of x does this change in sign occur?
• The acceleration of P always points directly downward and with a constant magnitude. Do you see this in the animation? And, do you see this in your analysis?

As we will see in the next class period when we work with the "path description" of kinematics, the velocity of a point is always tangent to the path of the point, and the acceleration always points "inward" on the path of the point. Do you see this in the animation below?