Solving Rotating Slotted Arm Problem

[Benny]

Hi, I'm not able to work through part of a question and I would like some help with it.

Firstly, I apologise for the lack of a diagram. The situation is this: First draw up the usual XY axes. A slotted arm (for the diagram it can be considered as a hollow retangular prism) has one end fixed at the origin. The length of the slotted arm is L. It rotates about the horizontal Z axis and the absolute angular velocity of the slotted arm is omega which is constant. So the slotted arm's angular displacement is wt.

Inside the hollow region on the slotted arm is a 'slider' (basically a small block) of mass m which is attached to a straight string. What happens is that as the string gets pulled, the slider moves in a straight line along the slotted arm.

The relative velocity v of the slider with respect to the arm is constant. The system of coordinates xyz is rigidly attached to the arm (ie. rotates with the arm). Note that the usual inertial coordinate system is XYZ, not xyz which is rotating.

Produce the expression for the components of the absolute linear velocity of the slider along the system of coordinates xyz.

Answer: $$\mathop {v_S }\limits^ \to = \left( { - v} \right)\mathop i\limits^ \to + \omega \left( {L - vt} \right)\mathop j\limits^ \to$$

I just can't figure out a way to get to the answer. I started by trying to find an expression for the absolute displacement of the slider, then differentiating to find the absolute velocity. I obtained an expression the component of the absolute velocity along the xyz system by 'dotting' the expression for the absolute velocity with i,j and k. Where I,J,K are the vectors corresponding to XYZ and i,j,k correspond to the xyz system.

The actual expression that I obtained for the absolute displacement of the slider is: $$\mathop R\limits^ \to = \left( {L - vt} \right)\cos \left( {\omega t} \right)\mathop I\limits^ \to + \left( {L - vt} \right)\sin \left( {\omega t} \right)\mathop J\limits^ \to$$

As explained before, when I differentiate this (L and v are both constant) and carry out the various manipulations I don't get the required result. I don't know what I'm doing wrong. Can someone help me out? Thanks.

Edit: Nevermind, I managed to get the x-component. Hopefully the other one will be easy to obtain. My error was in computing the dot product, leaving out half one of the expressions.

 

[member 553083]

The expression for the absolute velocity of the slider is \mathop {V_S }\limits^ \to = \mathop {\dot R}\limits^ \to = \left( { - v} \right)\mathop I\limits^ \to + \omega \left( {L - vt} \right)\mathop J\limits^ \to Taking the dot product of this with i,j,k, we get the required expression.