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The Physics Teacher ◆ Vol. 50, 2012
Physics Challenge for Teachers and Students
Boris Korsunsky, Column EditorWeston High School, Weston, MA 02493 [email protected]
Solution to November 2012 Challenge
w The Pin and the Pendulum
A particle attached to the end of a light string of length L forms a simple pendulum. A horizontal pin is located an unknown distance d directly underneath the pivot point. The particle is held so that the string is taut and horizontal and then released. What is the minimum distance d that would allow the string to wrap around the pin at least once?
(Submitted by John Mallinckrodt, Cal Poly Pomona, Pomona, CA)
Solution: The particle of the pendulum initially moves in a circle of radius L. When the string encounters the pin, the radius changes to L-D until the particle reaches the angle α. At this point (point 1) the particle begins to follow a parabolic trajectory.
At point 1, the particle is still in circular motion with instantaneous velocity v and the radius L-D. The par-ticle has a centripetal acceleration provided by the grav-ity.
2
csin( ) sin( ) vg a gL D
α α= → =−
2(1) sin( ) ( ).v g L - Dα→ = Due to the energy conservation the velocity v is:
2(2) 2 ( )
(3) ( ) sin( ).v g D hh L D α
→ = −→ = −
Combining (2) and (3): 2(4) 2 [ (1 sin( )) sin( )].v g D Lα α→ = + −
Now → (1) and (4): 3 sin( )(5) .
2 3sin( )LD α
α→ =
+
At point 1, the particle begins to follow a parabolic trajec-tory with origin (0,0).
Using the kinematic equations of parabolic trajectory:
2
(6) sin( )
(7) cos( ) .2
x v tgty v t
α
α
→ =
→ = −
The parabolic trajectory of the particle intersects the pin at the point (d,-h).
0
0
(8) 0 ( )cos( )
(9) 0 ( )sin( ).f
f
x x d L Dy y h L D
αα
→ = → = = −
→ = → =− =− −
Combining Eqs. (7) and (9): 2
cos( ) 0.2gt v hα− − =
Solving the quadratic equation, we obtain the time t when the particle of the pendulum passes by the pin.
2 2cos( ) cos ( ) 2(10) .v v ghtg
α α+ +→ =
Combining Eqs. (6) and (8): ( )cos( ) sin( ) .L D v tα α− =
Replacing the time t of Eq. (10) to (11):
D
L-D
H
L
Point
h
d
v
(d, -h)
α
Point 1
d
( )2 2sin( )( )cos( ) cos( ) cos ( ) 2 .vL D v v ghg
αα α α− = + +
Substituting Eqs. (4) and (5) to (12) and simplifying, we have: α < 36o.Replacing the angle α in Eq. (5) we obtain the mini-mum distance D that allows the string to wrap around the pin once: D @ 0.47 L.
(Contributed by Miguel Ángel Prieto Suárez, student, University of Seville, Seville, Spain)
We would also like to recognize the following successful contributors:
Don Easton (Lacombe, Alberta, Canada)Norge Cruz Hernández (University of Seville, Seville,
Spain)Jarrett L. Lancaster (James Madison University,
Harrisonburg, VA)Matthew W. Milligan (Farragut High School, Knoxville,
TN)Miguel Serrano Martín, student, (University of Seville,
Seville, Spain)Francisco Javier Tinoco Begines, student (University of
Seville, Seville, Spain)Sergio Toledo Jiménez, student (University of Seville,
Seville, Spain)
Guidelines for contributors:
– We ask that all solutions, preferably in Word for-mat, be submitted to the dedicated email address [email protected]. Each message will receive an automatic acknowledgment.
– The subject line of each message should be the same as the name of the solution file (see the in-structions below).
– The deadline for submitting the solutions is the last day of the corresponding month.
– We can no longer guarantee that we’ll publish every successful solver’s name; each month, a representative selection of names will be pub-lished, perhaps 10 to 15, both in print and on the web.
– If your name is—for instance—Joe Biden, please name the file “Feb13Biden” (do not include your first initial) when submitting the February solu-tion.
– If you have a message for the Column Editor, you may contact him at [email protected]; however, please do not send your solutions to this address.
As always, we look forward to your contributions and hope that they will include not only solu-tions but also your own Challenges that you wish to submit for the column.
Many thanks to all contributors and we hope to hear from many more of you in the future!
Boris Korsunsky, Column Editor
