HINTS FOR PROBLEM SET #18

Problem Set #18 – Problem 1

This problem involves looking at the motion (in this case just the velocity and not the acceleration) of a body (the collar) relative to a frame that is rotating (rod DB) relative to the fixed frame (the wall). 

So choose the rotating frame to be one which is rotating with rod DB. 

Point P (the collar attached to rod AP) is moving relative to this rotating frame. 

Part (a) is asking for the angular velocity of rod AP

Part (b) of this problem is simply asking for vP/DB.

Start this problem by writing out the equation that relates the absolute velocity of a point to its velocity relative to a rotating frame:

              vP = vP' + vP/DB

Let’s consider this term by term.

vP is the absolute velocity of P.  P, in its absolute motion, is rotating about A. 

Hence:

              vP = (omega of AP) x rP/A

We know rP/A.  We don’t know (omega of AP).  This is one of the quantities that we are looking for.

vP' is the absolute velocity of a point “glued” to the rotating frame and coinciding with point P.

Since we know the angular velocity of this frame and where P is, we can calculate this quantity.

Hence:

              vP' = (omega of DB) x rP/B

Lastly consider vP/DB.  We don’t know its magnitude but we do know the direction.

It must be directed along the rod DB (since the point P is on a collar on this rod). 

              vP/DB  = vP/DB(-cos40 i + sin40 j)

Put this all into the above equation and solve for the magnitude of vP/DB and the magnitude of the angular velocity of AP.  (Either graphically or analytically.)

Problem Set #18 – Problem 2 

Here is another problem that considers the motion (in this case just the acceleration) of a body moving relative to a frame that is rotating relative to a fixed frame. 

Point D is moving relative to the rod AB which is rotating relative to the fixed frame. 

Choose the rotating frame to be a frame rotating with rod AB. 

Point D is the point whose absolute acceleration is being asked for. 

Don’t let the fact that D is rotating confuse you.  The important thing is that D is moving relative to a frame that is rotating relative to the earth.  This is an example of when it can be a bit challenging to identify which frame you should choose to be the one rotating relative to the fixed frame.

Start this problem by writing out the equation that relates the absolute acceleration of a point to its acceleration relative to a rotating frame:

              aD = aD' + aD/AB + aC

First find aD'. 

This is the acceleration of a point D' coinciding with point D and rotating about a fixed axis at A.  It will have the angular velocity and angular acceleration of rod AB.

Next find aD/AB. 

This is the acceleration of a point D rotating about a fixed axis through B. 

It will have the angular velocity and angular acceleration of rod BD.

Lastly you need to get aC.  Recall that this Coriolis acceleration is twice the cross product of the angular velocity of the rotating frame (which is given) with vD/AB.  So you will need to get vD/AB.  Easy. It is just the velocity of a point rotating about a fixed axis at B with the angular velocity of rod BD.

Now plug all of this into the above and calculate aD.

Problem Set #18 – Problem 3 

Here again is another problem that considers the motion of a body moving relative to a frame that is rotating relative to a fixed frame.

The body is the pin D.  The rotating frame is rod AB.  D is moving relative to the rotating frame.  (Albeit in a hard to visualize manner.)

So start this problem by writing out the equation that relates the absolute acceleration of a point to its acceleration relative to a rotating frame:

              aD = aD' + aD/AB + aC

Let’s examine this equation term by term.

First consider aD.  You know the direction but not its magnitude. 

It has to be in the direction of the slot cut in the plate.

              aD = aD(-cos30 + sin30 j)

Next consider aD'.  It is just the acceleration of a point rotating about a fixed axis at A with the angular acceleration and angular velocity of AB.

              aD' = (alpha x rD/A) + (omega x (omega x rD/A))

where the alpha and the omega are for the rotating frame

You know the direction of aD/AB but not its magnitude. (It has to be in the direction of the slot cut in the rod AB.)

What about aC?  We can actually determine it completely, albeit with a lot of work.  Recall that aC is twice the cross product of the angular velocity of the rotating frame (which is given) with vD/AB. We just need to get vD/AB.  Use the equation that relates the absolute velocity of a point to its velocity relative to a rotating frame:

              vD = vD' + vD/AB

We can get vD' completely. (It is just the velocity of a point rotating about a fixed axis at A with angular velocity of AB.)

We know the directions of vD and vD/AB, but not their magnitudes. 

This is enough to solve for vD/AB.

Now use this to get the Coriolis acceleration.

Now plug all this into the equation that we stated with.  (You know…. The one that relates the absolute acceleration of a point to its acceleration relative to a rotating frame.) 

I suggest you solve the equation analytically since it is quite challenging to do it graphically (but not impossible.)