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DISCUSSION and HINTS
The animation below shows the motion of the disk as it moves along the incline. Included in the video are the friction and normal forces (FF and FN) acting on the disk as it moves.
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: vO and ωdisk.
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:
vO= vC + ωdisk x rO/C = ωdisk x rO/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.