Olympiad problems

2013 IPhOO, 10

Two masses are connected with spring constant $k$. The masses have magnitudes $m$ and $M$. The center-of-mass of the system is fixed. If $ k = \text {100 N/m} $ and $m=\dfrac{1}{2}M=\text{1 kg}$, let the ground state energy of the system be $E$. If $E$ can be expressed in the form $ a \times 10^p $ eV (electron-volts), find the ordered pair $(a,p)$, where $ 0 < a < 10 $, and it is rounded to the nearest positive integer and $p$ is an integer. For example, $ 4.2 \times 10^7 $ should be expressed as $(4,7)$. [i](Trung Phan, 10 points)[/i]

2017 F = ma, 23

23) A spring has a length of 1.0 meter when there is no tension on it. The spring is then stretched between two points 10 meters apart. A wave pulse travels between the two end points in the spring in a time of 1.0 seconds. The spring is now stretched between two points that are 20 meters apart. The new time it takes for a wave pulse to travel between the ends of the spring is closest to which of the following? A) 0.5 seconds B) 0.7 seconds C) 1 second D) 1.4 seconds E) 2 seconds

springs algebra

2017 F = ma, 2

2) A mass $m$ hangs from a massless spring connected to the roof of a box of mass $M$. When the box is held stationary, the mass–spring system oscillates vertically with angular frequency $\omega$. If the box is dropped and falls freely under gravity, how will the angular frequency change? A) $\omega$ will be unchanged B) $\omega$ will increase C) $\omega$ will decrease D) Oscillations are impossible under these conditions. E) $\omega$ will either increase or decrease depending on the values of $M$ and $m$.

springs algebra