Any questions??

Discussion and hints:

Your first decision on this problem is to choose your observer. Since an observer on the plate will have the simplest view of the motion of the insect, attaching the observer to the plate is recommended. Also, attach your xyz-axes to the plate.

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

Following that, determine the motion of the insect as seen by the observer on the plate.

Use these results with the moving reference frame kinematics equation to determine the velocity and acceleration of the insect.

Any questions??

Discussion and hints:

It is recommended that you use an observer attached to the wheel. As we have discussed in class, your choice of observer directly affects four terms in the acceleration equation: ω and α  (how the observer moves), and the relative velocity and relative acceleration terms (what the observer sees). Note that the remainder of the discussion here is based on having the observer attached to the wheel.

The wheel shown above has TWO components of rotation:

• a rotation rate of ω₁ about a fixed axis (the “+” Y-axis), and,
• a rotation rate of ω₂ about a moving axis (the “+” z-axis)

(Be sure to make a clear distinction between the lower case and upper case symbols.)

Therefore, the angular velocity of the wheel is given by:

ω = ω₁J + ω₂ k

The angular acceleration vector α is simply the time derivative of the angular velocity vector ω : α = dω/dt. In taking this time derivative,

• Recall that the J-axis is fixed. Since J is fixed, then dJ/dt = 0.
• Recall that the k-axis is NOT fixed. Knowing that, how do you find dk/dt?

With the observer attached to the wheel, what motion does the observer see for points A and B? That is, what are (v_A/O)_rel and (a_A/O)_rel, and (v_B/O)_rel and (a_B/O)_rel?

NOTE: Pay particular attention to the motion of the reference point O. What path does O follow? And, based on that, how do you write down the acceleration vector of O, a_O?

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

Any questions??

Discussion and hints:

Note that since the top contact point of the disk rolls without slipping on the upper fixed surface, this contact point has zero velocity. In addition, the horizontal component of acceleration of that point is also zero (the vertical component of acceleration is NOT zero, however).

For the velocity problem, write down the velocity kinematics equation for the disk (where “C” is the contact point):

v_O = v_C + ω x r_O/C = ω x r_O/C

and solve for the angular velocity of the disk. For the acceleration problem, write down the acceleration kinematics equation for the disk:

a_O = a_C + α x r_O/C – ω²r_O/C = a_C j + α x r_O/C – ω²r_O/C

and solve for the angular acceleration of the disk.

Then, write down and use the acceleration rigid body kinematics equation relating points O and B.

Carefully study the velocity (BLUE) and acceleration (RED) information for the point on the circumference of the disk shown in the animation below. Recall that the velocity of the point on the disk in contact with the fixed upper surface is zero, and its acceleration has only a vertical component. Do you see this in the animation?

Any questions??

Any questions??

Ask your questions here. And, consider answering questions of your colleagues here. Either way, you can learn.

DISCUSSION and HINTS

In this problem, end A of the bar is constrained to move along a straight horizontal path with a constant speed of v_A, whereas end B is constrained to move along a straight, angled path. As you can see in the animation below of the motion of the bar, the speed of B is NOT a constant (the acceleration of B is non-zero, and is, in fact, increasing as B moves along its path).

In your solution, it is recommended that you use the rigid body kinematics equations relating the motion of ends A and B:

v_B = v_A + ω x r_B/A
a_B = a_A + α x r_B/A – ω²r_B/A

For these equations, you know: i) the magnitude and direction for the velocity of A; ii) that the acceleration of A is zero (constant speed along a straight path); and, iii) the direction for the velocity and acceleration of B. These two vector equations produce four scalar equations that can be solved for four scalar unknowns: v_B, a_B, ω and α.

Discussion and hints:

The solution for this problem requires only a straight-forward application of the rigid body acceleration equation, relating the known accelerations of points A and C:

a_C = a_A + α x r_C/A – ω²r_C/A

Since this vector equation has two components (x and y), you have two scalar equations from which you can solve for ω and α.

Any questions??

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

Discussion and hints:

Enlarged view

This problem is  a straight-forward application of the planar rigid body kinematics equations. The velocity and acceleration of the center of the disk are known, as well as the angular velocity and angular acceleration of this disk. To find the velocity and acceleration of point A on the disk, you can use the following:

v_A = v_O + omega x r_A/O
a_A = a_O + alpha x r_A/O – omega^2*r_A/O

Using the enlarged view above, we can see the interplay between velocity (blue) and acceleration (red) of point A. Using what we learned back in Chapter 1, can you identify times when the speed of A is increasing and when it is decreasing? (Hint: Look at the angle between the velocity and acceleration vectors.) Using the top view, we can see the path of point A – is that the path that you expected for A?