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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)
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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.

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.

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.

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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