Answer to the Question 09/05

BREAKING A THREAD

The question was:

A weight is hanging on an elastic thread. An additional stretching force F is applied and is gradually (slowly) increased. When the force reaches value Fo the thread breaks. What should be the minimal size of a force that breaks the thread, if such a force is applied instantaneously and remains unchanged.

(1/06) This problem has been solved correctly (13/9/05) by Qiu Shi Wang and Ying Cun Luo, freshmen at Peking University, China (e-mail inklings@163.com), and (13/9/05) by Chetan Mandayam Nayakar, a student at India Institute of Technology, Madras, India (e-mail mn_chetan@yahoo.com). There are many, essentially equivalent ways to solve the problem. We will present what appears to be the simplest solution.


The answer: The thread will break if F=Fo/2.

The solution:
Before the force is applied the weight of the object hanging on the thread is balanced by the tension force of the thread. Once the additional force F is applied downwards the TOTAL force becomes F, and the weight starts executing harmonic oscillation under the influence of the forces. It starts the oscillation at the top point of the period. After a quarter of the period it reaches the midpoint of the oscillation at which the total force vanishes. After half of the period it reaches the bottom point of the oscillation, at which, by symmetry, the total force is F UPWARDS. This total force is result of the applied external force F pointing downwards, and the increase in the thread tension, which must be 2F and point upwards. Thus, the maximal thread tension is TWICE larger than the applied force. Consequently, F=Fo/2 suffices to break the thread.


Comment: This problem has been taken from an old issue of the journal "Kvant" ("Quantum") (Problem F1209, the issue of 1990). Copies of the old issues of that journal can still be found (in Russian) on the web. It is probably the best Physics-Mathematics journal for high-school children (as well as university students). It's a shame that nobody ever attempted to translate the wanderful collection of the problems that apeared there into English.
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