Zhiming Darren TAN

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FEYNMAN CRANK This problem in mechanics is well-discussed in Feynman’s book as an example to build physical intuition. The key is to realise that each arm can only exert forces parallel to their length because the pivots are frictionless. This can be proven by taking moments about the pivots. The other important element to keep clear in your mind is the distinction between velocity and acceleration. 1(a) $$F = {2} }{h}$$ Some failed to notice the constant L introduced in the text of the problem, and left answers in terms of angles. Some careless mistakes in the trigonometry and algebra were also observed. 1(b)(i) $$\tfrac12 u$$ By symmetry, the block will move with half the speed of the wheel. 1(b)(ii) The block _accelerates_ vertically. The vertical _velocity_ is upwards, but the _acceleration_ is downwards. You can get an intuition about this by thinking about how quickly the (vertical) velocity of the block changes when the arms are almost horizontal versus when the arms are almost vertical. Many explanations about the difference in the force were not fully satisfactory, as the link was not made specifically to _acceleration_ rather than to velocity. 1(b)(iii) $$v = {2h}$$ The actual _velocity_ is in the tangential direction, so the expression is most easily obtained by considering that the horizontal velocity ${2}$ is a _component_ of the tangential velocity. The angles are a bit tricky, especially if you do not draw your own diagram. The diagram provided has the angles close to π/4, which can lead to confusion between sines and cosines. 1(b)(iv) $$a_{radial} = {L} = {4h^2}$$ Since we know that the actual _acceleration_ is in the vertical direction from 1(b)(ii), this radial acceleration is a _component_ of the acceleration. $$a = a_{radial} {h} = {4h^3}$$ 1(b)(v) $$F = {2} \left( g - {4h^3} \right) }{h}$$ Surprisingly many were mixed up about the signs in this last part. The acceleration of the block should be linked to the resultant force on the block, and the weight of the block is _opposite in direction_ to the upward force from the arms. _[The original question I had in mind would have gone on to energy considerations, but this was omitted to keep the test shorter.]_
A1

Zhiming Darren TAN

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FRAMES OF REFERENCE Colloquially, a frame of reference refers to one’s view of the world. The term “worldview” is a calque from the German “Weltanschauung”. In physics, a frame of reference (or reference frame) is a SYSTEM OF COORDINATES FOR SPACE AND TIME in order to describe the natural world. You need a reference frame before you can discuss physics, even if you are not explicitly aware of this! Different observers can have different reference frames. Gallilean Transformations Newton’s laws of motion relate forces to acceleration. These laws are thus valid for any observer who is moving with constant velocity relative to distant “fixed” stars, since there is NO ACCELERATION INTRODUCED by the choice of reference frame. We call such reference frames “inertial” frames of reference, and these inertial frames are related to any other inertial frame by the ADDITION OF A CONSTANT VELOCITY VECTOR to all the velocities measured in the other frame. (For simplicity, we are ignoring more trivial ways in which reference frames can differ, e.g. in the relative orientation of the direction for “up” or “right”, the choice of units for measuring distance and time.) This addition of a constant velocity is what we term a Gallilean transformation, since Gallileo first described this principle of the invariance of mechanical laws in 1632 using the example of a ship travelling on a smooth sea . We also describe such switches between reference frames as “boosts” from one frame to another. Suppose the inertial frame S′ is moving with velocity $$ relative to the inertial frame S, and the origin of both frames coincide at time t = 0 = t′. Mathematically, the GALLILEAN TRANSFORMATION for position is: $$ = - t$$ where t = t′ is the time measured by the common synchronised clock in both frames. There is a NEGATIVE SIGN in this equation because the frame S′ is moving with velocity $$ as seen by S. (Note that Einstein realised that such a concept of “absolute time” is not tenable, and developed relativity theory, but we won’t go into that here.) The velocity comes from the time derivative of position, $$ = }{dt'} = }{dt} - $$ and if we differentiate once more with respect to time to get acceleration, we find no difference between the accelerations measured in either frame.
INTRODUCTION Four runs of this 2-day teacher workshop were held between June 2017 and November 2018, facilitated by Kwek Leong Chuan. Quite a lot of relevant resources for teaching and learning can be found here (including a link to the latest version of this document). A huge collection of simulations is hosted here. Main Points 1. Encourage greater independence and exploration by students, encourage their intellectual curiosity and awesomeness. Help nurture their “taste” for beauty in physics. 2. Read the book by Gilbert Highet, _The Art of Teaching_. Learning, thinking, exploring... the path to happiness! 3. Informal learning experiences beyond the classroom will be important to complement syllabus-oriented instruction. These can include going for a symposium or colloquium talk, or even mini-projects and explorations. 4. H3 Physics should prepare students well for a range of future pursuits, be they in science, engineering, mathematics or even politics and economics. 5. A great variety of teaching and Learning (T&L) activities can be used. In physics education, we have the advantage of bringing in demonstrations (e.g. van de Graff generator) and experiments, and for mechanics there are many little toys that illustrate very interesting phenomena. Students can tinker around with breadboards and circuit components for electronics. Many videos and online resources like simulations are readily available, so students can learn the content on their own (flipped classroom). The physics education research literature is also quite rich, with education-focused journal articles suggesting ideas for curriculum delivery, documenting common student misconceptions, and exploring interesting physics phenomena from a pedagogical angle. 6. Students should not fear mathematics. Logic is essential in reasoning about patterns and relationships between variables. Basic techniques like dimensional analysis are very powerful.