Re: Free Body Diagram

[John Clement <clement@HAL-PC.ORG>]

> In Giancoli chapter 6 (Energy and Work) there is a problem wherein a
> person runs towards and grabs a vertical hanging rope in order to swing
> out over the water of a lake and then release the rope to fall into the
> water.  The question asks for the tension in the rope at the point in
> his swing where his velocity is zero.
>
> Now, it seems to me that one could look at the free body diagram at that
> instant and it would show the weight of the person acting down and the
> tension acting at an angle theta from the vertical (upward, at an angle
> from the vertical).
>
> If one then considers a coordinate system imposed at the end of the
> rope, one might argue that Tcos(theta) = mg in order that there is no
> vertical acceleration.
>
> On the other hand, if one uses a rotated coordinate system (with T along
> the y axis), one could argue that T=mgcos(theta) in order that there no
> acceleration along the direction of the rope.
>
> Only one of these points of view can be correct and I am searching for
> the words that my high school class will understand and will convince
> them (and me) why one is correct and the other incorrect.
The idea that the vertical acceleration goes to zero is actually a variant
of the common misconception that students have.  They think that the
acceleration goes to zero when a ball is thrown up and reaches it's peak
where the velocity is zero.  Students may wish to hang onto that explanation
as the misconception is fairly resistant to change.

Of course Bob's explanatons are correct, but it may be very unconvincing to
students who still harbor the previous misconception.  However students who
do not harbor the previous misconception might be able to draw the correct
free body diagram.  At the point of zero velocity, the change in velocity
can not have a radial component, so the tension + gravity must have a
tangential resultant.  If there were a radial component, the person would
acquire a radial velocity.  This argument of course only works at zero
velocity, as a radial force component is necessary to change the direction
of a non zero velocity.

John M. Clement
Houston, TX

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.