Teodora Č.

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Physics (Newtonian Mechanics)

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Question:

If we have a body of mass $$m$$ and it circles around point $$p$$ with constant velocity $$v = const.$$. Is frame attached to a body an inertial frame?

Teodora Č.

Answer:

Inertial coordinate systems in Newtonian mechanics are ones where the body in the system that is experiencing acceleration which is provided only by interaction with other bodies in that system, by Newton's laws. That means, according to Newton's second law, that an inertial system must be $$\textbf{static}$$ or moving with $$\textbf{constant}$$ velocity relative another inertial frame. We have that body is moving at constant speed that means $$|v| = const$$, but $$\frac{d\vec{r}}{dt} = \frac{dr}{dt}\vec{e_{\theta}} + \frac{d\vec{e_{r}}}{dt}r$$ while $$ \frac{dv}{dt}\vec{e_r} = 0$$ the change in direction of $$\vec{r}$$ vector is $$\frac{\vec{de_r}}{dt}v \neq 0$$. When we tako the derivative again we get: $$\vec{a} = r\frac{d\omega}{dt} \vec{e_{\theta}}(t) - {\omega}^2r \vec{e_r}(t)$$ since $$r\frac{d\omega}{dt} = \frac{d(\omega r)}{dt} = \frac{d|\vec{v}|}{dt} = 0$$ in our case and we have resulting acceleration: $$\vec{a}(t) = -{\omega}^2r \vec{e_r}(t)$$ So we have resulting non-zero acceleration, that means according to Newton's second law that the body in that frame is experiencing a force. And the force that body in this (frame attached to the body) frame experiences due to rotation of that frame is exactly: $$\vec{F} = m\vec{a} = -m{\omega}^2 r\vec{e_r}$$ So we conclude that frame attached to body in uniform circular motion is $$\textbf{not inertial}$$

Basic Math

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Question:

What's the difference between sequence and sum?

Teodora Č.

Answer:

A sequence is just list of numbers or values that have a specific place in order. For example: $$x_n = {2, 4, 6, ... 2n}$$ which is the sum of all even integers starting from $$1$$ to $$n$$. And these numbers are ordered like this: - the first member is 2 - the second member is 4 - the third member is 6 ... So we see that every member of a sequence has a place in a sequence like this. Sum is when we take sum of all sequence members. Sum of the first sequence we mentioned $$x_n$$ will be: $$\sum{x_n} = 2 + 4 + 6 +...+2n$$ And if we talk about finite sum, for example from n=1 to 5, which means we are going to sum all members of a sequence from the first to the fifth member then we write: $$\sum\limits_{i=1}^5 x_n = 2 + 4 + 6 + 8 + 10 = 30$$ and we call this $$\textbf{partial sum}$$ from $$1$$ to $$5$$, where $$1$$ means first member, and $$5$$ means fifth member of a $$x_n$$ sequence.

Astrophysics

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Question:

Try to approximate wavelength of your own body radiation, if you assume you radiate black body radiation. Assuming temperature of the Sun is $$T_{sun} = 5700 K$$ and $$T_b$$ in Kelvins, as your own (about $$305K$$ ).

Teodora Č.

Answer:

If you use Wien's displacement law then you approximate wavelength at which temperature is at its maximum $$\lambda_{max} = \frac{b}{T_{sun}}$$, according to Plank's law of radiation. Then as you approximate yourself as a black body, you use Wien's displacement law on yourself as well. $$\frac{\lambda_{{max}_b}}{\lambda_{{max}_{s}}} = \frac{T_{sun}}{T_{b}}$$ And as you know, you can also assume that maximum of Sun's radiation is in visible part of electromagnetic spectrum, therefore, as visible electromagnetic spectrum range is from $$700nm$$ to $$380nm$$, you can use average from that ranges, which is $$\overline{\lambda}_{max} = 540 nm$$. Then you have $${\lambda_{{max}_b}} = \frac{T_{sun}}{T_{b}}\lambda_{{max}_{s}}$$ then when you use Wien's law to human-body emission results in a peak wavelength of $${\lambda_{{max}_b}} = \frac{5700K}{310K}540\cdot10^{-9}m = 940\mu m$$ Which is in $$\textbf{infrared}$$ range of electromagnetic spectrum spectrum.

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