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In the animation of the simulation shown below, the RED vectors shown are the forces of reaction acting on particles A and B (such as the force on each particle by member AB, and the normal forces of reaction by the floor and wall).

Use the Four-Step solution plan outlined in the lecture book:

Step 1 - FBD: Draw individual FBDs of blocks A and B.

Step 2 - Kinetics (Newton): Write down the Newton's 2nd Law equations for A and B from your FBDs above.

Step 3 - Kinematics: This part of the problem solution requires the most thought. Suppose that you have an observer that is moving along with block B. What is the direction of the motion of A that is seen by this observer. (HINT: The observer sees A moving in the direction along the inclined interface of the two blocks.) Using this, you can write:
a_A = a_B + a_A/B
or,
a_A*j = a_B*i + a_A/B*(cos(theta)*i + sin(theta)*j)
Use the two components of this vector equation to relate a_A and a_B.

Step 4 - Solve.

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In order to draw the FBDs for the case with friction, you need to know first the direction of impending motion; that is, you need determine whether A moves UP the incline or DOWN the incline, since friction will oppose that motion. To determine the direction of impending motion, solve the problem for the friction-free case first (μ_k = 0). Once this direction is determined, rework the problem with friction.

Recall the following four-step plan outline in the lecture book and discussed in lecture. You will need to do this for both the friction-free case and the with-friction case.

Step 1: FBDs
Draw individual free body diagrams for the two blocks.

Step 2: Kinetics (Newton's 2nd Law)
Write down Newton's 2nd law for each block.

Step 3: Kinematics
Recall the work that we did in Chapter 1 in relating accelerations of two particles connected by a cable-pulley system.

Step 4: Solve
At this point you will have two equations in terms of two unknowns: the normal contact force and the tension force. Solve.

Ask and answer questions here. You learn both ways.

Ask and answer questions here. You learn both ways.

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) for the car. What is the difference between the FBDs for Parts a) and b)? They are not the same!

Step 2: Kinetics (Newton's 2nd Law)
Since the position of the car is known in terms of its path, a path description for resolving the forces acting on the car is recommended. What directions are e_t and e_n for this problem? You will need to use a third unit vector k for the vertical direction.

Step 3: Kinematics
Since you have used a path description for resolving forces, you should also use a path description for acceleration; that is, a_P = v_dot*e_t +(v²/ρ) e_n. 

Step 4: Solve
At this point you will have two scalar equations to solve.

Ask and answer questions here. You learn both ways.

DISCUSSION and HINTS

Shown below is an animation of the motion for particle P as it moves around on its circular path. Note that the normal force acting on P switches from contact on the outer surface (normal force pointing inward) to contact on the inner surface (normal force pointing outward) and back to contact on the outer surface. Also note that the friction force always opposes the sliding motion of P, as should be expected. (Note that the parameters used in the simulation producing this animation may differ from the ones used in this semester's homework set.)

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

Step 1: FBDs
Draw a free body diagrams for P. Be sure to show the friction force opposing the motion of P in your FBD.

Step 2: Kinetics (Newton's 2nd Law)
Using a set of path unit vectors for resolving forces into components is recommended here. Write down the appropriate Newton's 2nd law equation in normal and tangential directions. Recall that f = μ_k*|N|.

Step 3: Kinematics
Since you have used a path description for resolving forces, you should also use a path description for acceleration; that is, a_P = v_dot e_t +(v²/r) e_n.

Step 4: Solve
At this point you will have two equations in terms of two unknowns: N and v_dot.

Ask and answer questions here. You learn both ways.

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 for the block.

Step 2: Kinetics (Newton's 2nd Law)
Since the position of the block is given in terms of a radial distance r and a rotation variable θ, a polar description is recommended for resolving the forces acting on the block.

Step 3: Kinematics
Since you have used a polar description for resolving forces, you should also use a polar description for acceleration; that is, a_P = (r_ddot - r*θ_dot²)e_r +(r*θ_ddot + 2r_dot*θ_dot) e_θ. What is true about r_dot and r_ddot when the cable remains taut?

Step 4: Solve
At this point you will have two equations in terms of two unknowns: the normal contact force and the tension force. Does your answer for the normal force on the block make sense?

Any questions??

Discussion and hints:

You need to write down the angular velocity and angular acceleration of the football. Based on what we have been doing up to this point in Chapter 3, hopefully it is clear that the football (and insect) has two components of angular velocity: ω₁ about the fixed X-axis and ω₂ about the moving x-axis. Take a time derivative of the angular velocity vector to find the angular acceleration of the football (observer).

Any questions??

Ask and answer questions here. And, learn from both.

DISCUSSION and HINTS

For your work on this problem, it is recommended that you use an observer attached to the disk. The observer/disk has two components of rotation:

• One component of ω₀ about the fixed K-axis.
• The second component of ω_disk about the moving j-axis.

Write out the angular velocity vector ω in terms of the two components described above.

Take a time derivative of ω to get the angular acceleration α of the observer/disk. When taking this derivative, you will need to find the time derivative of the unit vector j. How do you do this? Read back over Section 3.2 of the lecture book. There you will see: j_dot = ω x j, where ω is the total angular velocity vector of the disk that you found above.

Acceleration of point A
The motion of A is quite complicated. To better understand the motion of A, consider first the view of point A by our observer who is attached to the disk - what does this observer see in terms of relative velocity and relative acceleration: (v_A/B)_rel and (a_A/B)_rel?

With this known relative motion, we can use the moving reference frame acceleration equation:

a_A = a_B + (a_A/B)_rel +α x r_A/B + 2 ω x (v_A/B)_rel + ω x (ω x r_A/B)

In finding the acceleration of B, a_B, note that B moves with a constant speed on a circular path centered on point O. Use the path description to find a_B. WARNING: Although B moves with a constant speed, its acceleration is NOT zero.

Ask and answer questions here. And, learn from both.

CHANGE IN PROBLEM STATEMENT:  Please use θ = 0 = constant when solving this problem.

DISCUSSION and HINTS

For your work on this problem, it is recommended that you use an observer attached to the disk. The observer/disk has two components of rotation:

• One component of ω₀ about the fixed J-axis.
• The second component of ω₁ about the moving i-axis.

Write out the angular velocity vector ω in terms of the two components described above.

Take a time derivative of ω to get the angular acceleration α of the observer/disk. When taking this derivative, you will need to find the time derivative of the unit vector i. How do you do this? Read back over Section 3.2 of the lecture book. There you will see: i_dot = ω x i, where ω is the total angular velocity vector of the disk that you found above.

Velocty of point A
The motion of A is quite complicated. To better understand the motion of A, consider first the view of point A by our observer who is attached to the disk - what does this observer see in terms of relative velocity: (v_A/O)_rel?

With this known relative motion, we can use the moving reference frame velocity equation:

v_A = v_O + (v_A/O)_rel + ω x r_A/O