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Mykolas Z.
Online tutor for a year; taught undergraduates at Cambridge for four years
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Physics (Newtonian Mechanics)
TutorMe
Question:

An ice skater is spinning on ice at some constant angular velocity $$\omega$$ with their arms stretched out. When they pull their arms in while still spinning, $$\omega$$ increases. Explain why.

Mykolas Z.
Answer:

We apply conservation of angular momentum: $$I_1\omega_1 = I_2\omega_2$$. By pulling their arms in, the ice skater decreases their moment of inertia $$I$$. So if $$I_2 < I_1$$ and $$I_1\omega_1 = I_2\omega_2$$, it means that $$\omega_2 > \omega_1$$: their angular velocity has increased.

Calculus
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Question:

Integrate $$x \ln{(x)}$$.

Mykolas Z.
Answer:

We use integration by parts, which can be derived by differentiating a product of two functions: $$\int u dv = uv -\int v du$$. Applying this rule once, with $$v = \frac{x^{2}}{2}$$ and $$u = \ln{x}$$, we get: $$ \int x\ln{(x)} dx = [\frac{x^{2}}{2}\ln{(x)}] - \int{\frac{x^{2}}{2}\frac{1}{x}dx} + C_0 = [\frac{x^{2}}{2}\ln{(x)}] - \frac{x^{2}}{4} +C_1$$, where $$C_0$$ and $$C_1$$ are integration constants.

Physics
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Question:

You have three resistors, each having a resistance of 5 Ohms. Construct a circuit that has the lowest possible resistance using all three resistors. What is that resistance? Now construct a circuit using all three resistors that has the highest possible resistance. What is it?

Mykolas Z.
Answer:

The lowest resistance would be when the resistors are all connected in parallel. The resistance then becomes: $$\frac{1}{R} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}$$; $$R = \frac{5}{3} \Omega$$. The highest resistance is achieved when all three resistors are connected in series, $$R$$ then becomes: $$R = 5 + 5 + 5 = 15 \Omega$$.

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