USAPhO 1999, Четвъртфинал — Задача 2
Preliminary Exam · Open Response Questions
Solutions to Problems
Any correct, equivalent solution should be awarded proportionate points. You may further break down the listed points into one point increments. Students should not be penalized in a subsequent part for using the wrong answer to a previous part. (No double jeopardy.)
Условие
A cube of mass M starts at rest at point A at a height of 4R, where R is the radius of the circular part of the track. The cube slides down the track and around the loop. The cube is very small compared to the size of the track.
a. Assuming the track is frictionless, find the force the track exerts on the cube at point B. Express your answer as a function of Mg. [15 т.]
b. The cube is replaced by a sphere with mass M and radius r (). Assuming that the sphere starts at the same height 4R and rolls without slipping, find the force the track exerts on the sphere at point B. Express your answer as a function of Mg. [10 т.]
(contributed by Mary Mogge)
Отговори
a.
Directed downward.
b.
Directed downward.
Решение
- a. Apply energy conservation to equate total energy at point A and point B. [2]
[3]
[1]
Applying Newton's 2 law to the forces at point B [2]
[4]
[1]
or downward [2]
b. Since it is rolling at point B, it also has rotational kinetic energy there
[3]
With [3]
[1]
[1]
Thus downward [2]
Автори
The 1999 Preliminary Examination open response questions were written by the coaches of the United States Physics Team. The coaches are: Academic Director, Dr. Mary Mogge – Professor of Physics at California State Polytechnic University, Pomona, CA; Senior Coach, Dr. Leaf Turner – Physicist in the Theoretical Division of Los Alamos National Laboratory, Los Alamos, NM; Dr. Warren Turner – Physics Teacher at St. Paul's School, Concord, NH. The coaches would also like to thank former academic director, Dr. Larry Kirkpatrick for his many helpful comments.
Оригинал в Архива: Quarterfinal 1999.pdf · официални решения: Quarterfinal 1999 Solutions.pdf