7. Concluding Remarks
Classical Mechanics is one of the topics with the longest background in the physics olympiad. As such, though problems usually present themselves with a pretty straightforward approach, often times the trick lies in viewing questions with a different perspective (i.e. system versus components, choosing the correct frames, etc), or questioning basic assumptions (see supplementary question 1).
What is important, then, is to build up your mathematical foundations, and slowly expand your repository of skills. With each question solved, ask yourself if you could have done it in another method, if there was a different approach that might have worked. Do this well, do this consistently, and you will see improvements in your skills over time.
8. Supplementary Questions
8.1 Two bars of masses \(m_1\) and \(m_2\) connected by a non-deformed light spring of spring constant \(k\) rest on a horizontal plane. The coefficient of friction between the bars and the surface is \(k\). What minimum constant force has to be applied in the horizontal direction to the bar of mass \(m_1\) in order to shift the other bar. \((m_1 + m_2 /2) kg\)
8.2 A fly is able to provide itself a thrust \(F\) to fly in dense air which against an air drag \(-kv\). Assuming that it is able to fly at \(v_1\) upwards, \(v_2\) downwards, what is the maximum velocity that it can fly horizontally? \(\sqrt{v_1 v_2}\)
8.3 A projectile is launched at velocity \(v_0\) at an angle \(\theta\) above the horizontal. What should \(\theta\) be such that its trajectory length is maximum? Assume that the start and end points of the trajectory are at the same height. (approx 56.5 degrees)
8.4 A ball is thrown upwards with velocity \(v_0\).
8.4.1 Ignoring air drag, what is time taken for the ball to come back down to where it started?
8.4.2 Assuming that the air drag takes the form \(F=-m\alpha v\), where \(\alpha\) is a positive constant, solve for the maximum height of the ball.
8.4.3 Obtain an implicit equation for the speed of the ball when it comes back down, \(v_f\)
8.4.4 Determine the time taken for the ball to complete the entire journey. Is this longer or shorter as compared to the time taken for it to complete the same journey in the absence of air drag? (shorter)
8.5 A triangular prism of mass M is placed one side on a frictionless horizontal plane as shown below The other two sides are inclined with respect to the plane at angles and respectively. Two blocks of masses \(m_1\) and \(m_2\), connected by an inextensible thread, can slide without friction on the surface of the prism. The mass of the pulley, which supports the thread, is negligible.