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: [tex]\mathop {v_S }\limits^ \to = \left( { - v} \right)\mathop i\limits^ \to + \omega \left( {L - vt} \right)\mathop j\limits^ \to [/tex]

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: [tex]\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 [/tex]

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.

 

[Nate810]

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.

 

[quicknote]

Dear fellow scientist,

Thank you for reaching out for help with this problem. It seems like you have made significant progress in your calculations. However, it can be easy to make small errors in complex equations, so it's great that you were able to identify your mistake and correct it. Keep up the good work!

In terms of finding the other component of the absolute linear velocity, you may want to consider using the Pythagorean theorem to find the magnitude of the velocity vector, and then use trigonometric functions to find the angle between the velocity vector and the x-axis. From there, you can use basic trigonometry to find the components of the velocity vector along the x and y axes.

I hope this helps and good luck with your further calculations! If you encounter any other difficulties, don't hesitate to reach out for help. Remember, even the most experienced scientists make mistakes and need assistance from time to time. Keep persevering and you will find the solution.

Best regards,

Scientist