Newton's law of universal gravitation

Select the correct formula for Newton's law of universal gravitation.

There are two masses present, m1 and m2, the distance between their centres of gravity is r.

Select the correct option.

$$F_{grav}= {G\times m_1\times m_2\over r^2}$$, where G is the gravitational constant, which has the value $$6.67\times10^{-11}Nm^2/kg^2$$.

Gravitation

Which force exists between two ships with masses of $$m_1=500ton$$ and $$m_2=1000ton$$ when the distance between their centres of mass is 20m?

1 ton = 1000kg

$$F = {G\times m_1\times m_2\over r^2}={6.67\times10^{-11}\times500000\times100000\over 10^2}\approx0.03N$$

Distance of the Earth from the Sun

Calculate the average distance of the Earth from the Sun.

$$m_{Sun}\approx2\times10^{30}kg$$

$$m_{Earth}\approx5.97\times10^{24}kg$$

Gravity is the cause of the centripetal force, so these two forces are equal:

$${G\times m_{Earth}\times m_{Sun}\over r^2}={m_{Earth}\times v^2_{Earth}\over r}$$

The speed of the Earth can be expressed using the time it takes to orbit: $$v={2\pi r\over T}$$, so

$${G\times m_{Sun}\over r^2}={(2\pi r)^2\over T^2r}$$, therefore, $$r=({G\times m_{Sun}\times T^2\over(2\pi)^2})^{1/3}\approx 1.49\times 10^{11}m$$

Luminosity of the Sun

Sun has a radius of $$7.0 \times 10^8m$$ and a serface temperature of 6000K. Calculate the power output of the Sun.

Using the Stefan-Boltzmann law, $$W_{Sun}=\sigma\times4\pi r^2\times T^4 \approx 5.67\times10^{-8}\times 6.16\times10^{18}\times 6000^4 \approx4.52\times10^{26}W$$, where $$\sigma =5.67 \times 10^{-8}W m^2/K^4$$is Stefan-Boltzmann constant.

Temaperature of a star

Sun has a radius of $$7.0 \times 10^8 m$$ and a luminosity of $$4.52\times10^{26}W$$.Calculate the surface temperature of a star of radius $$20.0 \times 10^8 m$$ with the same luminosity as the Sun.

Using the Stefan-Boltzmann law, $$W_{Sun}=\sigma\times4\pi r^2\times T^4$$, where $$\sigma =5.67 \times 10^{-8}W m^2/K^4$$is Stefan-Boltzmann constant. Therefore, the surface temperature of the star is $$T=({W_{Sun}\over\sigma\times4 \pi r^2})^{1/4}\approx({4.52\times 10^{26}\over5.67\times10^{-8}\times 5.03\times10^{19}})^{1/4}\approx3548K$$

Wien law

The surface temperature pf P Cygni is 18700K. Calculate the peak wavelengh in the spectrum of this star.

By Wien's law, $$\lambda_{max}T=2.898\times 10^{-3}mK$$, so $$\lambda_{max}=2.898\times 10^{-3}mK/18700\approx1.55\times 10^{-7}m$$

Main sequence

The Hertzsprung-Russell diagram is shown on the picture below. Select the digit that shows the main sequence.

The main sequence is a continuous and distinctive band of stars that appears on plots of stellar color versus brightness.

White dwarfs

The Hertzsprung-Russell diagram is shown on the picture below. Select the digit that shows white dwarfs.

The white dwarfs are found in the bottom-left of Hertzsprung-Russell diagram. They lay below the main sequence.

Giants

The Hertzsprung-Russell diagram is shown on the picture below. Select the digit that shows giants.

Giants form a cloud above the main sequence, but they are not found in the upper part of Hertzsprung-Russell diagram.

Supergiants

The Hertzsprung-Russell diagram is shown on the picture below. Select the digit that shows supergiants.