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Re: Centripetal force and TPT article



Chris Horton wrote:

Teaching a conventional course, I must say I had the devil of a time trying
to get students to distinguish between an actual force (the force being
applied by a string and the force being applied by gravity) from the
centripetal component.

If you think your students have this figured out, give them a bob on a
string being swung in a vertical plane to solve and explain. I almost
guarantee a mish-mash.

I agree with John D - a lot of fuss over a small issue. Yes there can
be student difficulties, but it almost sounds like there are some
instructor difficulties. The following ideas might help mitigate some
of the difficulties.

Introduce the centripetal force in the same way you introduce say the
x-component of the net force. Emphasize that it's the same idea. We
have F-vector = m * a-vector. We can choose to resolve the vectors on
each side of this equation into rectangular components (x,y) or polar
components (r,theta). If we choose the origin to be at the
instantaneous center of curvature, then the polar components become
(centripetal,tangential).* [See footnote.] I suggest you introduce
the tangential component at the same time as you introduce the
centripetal component and always ask students about both in a given
problem. Many texts put the centripetal and tangential pieces into
different chapters, which isn't helpful when the concept of polar
components is first introduced in the course.

(As an aside, I find it helpful to do the same for the contact force
between two objects: emphasizing that we resolve it into two
components that we conventionally call normal and friction forces.
This can avoid difficulties in problems with say a wheel driving over
a curb: while it's driving over, which direction does the "normal"
force point? That's silly, it's much better to just mark a single
contact or reaction force in some arbitrary direction. We do the same
thing for example in the problem of a boom hinged to a vertical wall
and supported by a string connected to its far end. Marking a single
hinge reaction force in an arbitrary direction is generally easier
than trying to "guess" in advance which components must be zero.)

It might help if you have a policy of NEVER allowing students to put
force components on their free-body diagrams: F_x may not appear, nor
may F_r, F_t, or whatnot. BUT insist they always put on a coordinate
system and origin AND insist they always put one or more arrows
(distinguished in some clear way from the force arrows) to indicate
the net acceleration OR one complete set of components (but NOT both).

In your pendulum example given to my students, the FBD of the bob
would include the following:

- m (labeled mass of bob)
- mg and T (labeled arrows attached to bob)
- x = horizontal, y = vertically down, 0 at string attachment point
- a_c toward 0, a_t tangentially downward (labeled arrows not attached to bob)

I am very particular that my students have these four elements
(masses, forces, coordinate system, accelerations) clearly indicated
on their diagrams. I am always amazed by texts that don't put
accelerations, for example, on FBDs. Without this, how can you figure
out which of the following equations say is correct:

F1 - F2 = ma
F2 - F1 = ma

Carl

*Footnote: Textbooks irk me greatly when they state or imply that the
radial and centripetal components are synonyms, or similarly for the
azimuthal and tangential components. They're only the same thing for
a particle moving in a circular path with the origin at the center of
the circle. If you choose the origin at another point or if the path
is noncircular, then a_c is not equal to a_r, and a_t is not equal to
a_theta, in general. With nonmajors I don't worry about this too
much, but for majors the distinction is worth making, as it leads
into the standard elegant derivation of the Coriolis and centrifugal
forces:

http://physics.usna.edu/physics/faculty/mungan/Scholarship/CentripetalAcceleration.pdf

http://physics.usna.edu/physics/faculty/mungan/Scholarship/Acceleration.pdf
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)
U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu http://physics.usna.edu/physics/faculty/mungan/