Chapter 15 Problem 153

Using Aaron's Method to Solve the Problem

The following problem is solved using Aaron's Method. This method will be explained in detail in the method section of the solution. The method section is included in this solution to facilitate understanding of Aaron's Method (it is not used to actually solve the problem).

The Problem

Step-By-Step Solution

The images of the solution is shown below:

To gain and understanding of the coordinate system used and the ground point please see the explanation in the previous problem.

Hover over the curved arrows and ground point to see the text appear.

Recognize when to use a rotational coordinate system.

All previous relative velocities did not involve using a rotational reference frame. But in this case we have to use one because of the sliding collar. 

One way to tell when a rotating reference frame will be used is by the distance of points given in the problem. 

If the given distance of two points in the problem will be changing, then a rotating reference frame will most likely be used.

For this example the distance between the points EB is given and will be changing with time.

Whereas the distances for points AB and EF will be constant.

A way to visualize a rotational coordinate system.

For this problem there is a way to easily visualize v_B/E = v_B'/E + (v_B/E)_rot.

For v_B'/E this can be done by imagining that the collar B is removed and replaced with a hinge at the same position B'.

If this were the case then we would be dealing with our usually rigid bodies and the equation would simplify to v_B/E = omega_BE x r_E-->B.

But point B is not rigidly connected to point E, it is allowed to slide on the bar ED. To take account of this sliding we include another term.

The term (v_B/E)_rot means that we are looking at the velocity of the collar along the rod ED.

We do this by imagining the rod to be fixed and seeing how the collar moves along it, in this instant it will move side to side.

In summary v_B/E can be found by holding the collar fixed and moving the rod, v_B'/E, followed by holding the rod fixed and moving the collar, (v_B/E)_rot.

Choosing a path to find the answer.