Question

Mars has a mass of 6.42×1023 kg and a radius of 3.40×106 m. Assume a person is standing on a bathroom scale on the surface of Mars. Part A Over what time interval would Mars have to complete one rotation on its axis to make the bathroom scale have a zero reading? Express your answer with the appropriate units.

Answer 1

Answer:

T = 1.67 hours

Explanation:

It is given that,

Mass of mars, $M=6.42\times 10^{23}\ kg$

Radius of mars, $r=3.4\times 10^{6}\ m$  

Let m is the mass of the person standing on a bathroom scale on the surface of Mars. As the person rotates on its axis, the force of gravitation is balanced by the centripetal force as:

$m\omega^2 r=\dfrac{GmM}{r^2}$

$\omega^2 r=\dfrac{GM}{r^2}$

Since, $\omega=\dfrac{2\pi}{T}$

$(\dfrac{2\pi}{T})^2 r=\dfrac{GM}{r^2}$

$T^2=\dfrac{4\pi ^2r^3}{GM}$

$T^2=\dfrac{4\pi ^2\times (3.4\times 10^{6})^3}{6.67\times 10^{-11}\times 6.42\times 10^{23}}$

T = 6019.60 seconds

or

T = 1.67 hours

So, the time taken is 1.67 hours. Hence, this is the required solution.

Answer 2

The time interval that Mars would have to complete one rotation on its axis to make the bathroom scale have a zero reading is 1.67 Hours

Given the data in the question;

• Mass of mars; $M = 6.42*10^{23}kg$

• Radius of mass; $r = 3.40*10^{6}m$

Determine get the time interval that Mars would have to complete one rotation on its axis to make the bathroom scale have a zero reading.

Now, for mass to complete one rotation, it requires centripetal acceleration and this acceleration is provided by gravitational acceleration:

Hence, Centripetal acceleration = Gravitational acceleration

So we have;

$w^2 * r = \frac{GM}{r^2}$

We know that, angular velocity; $w = \frac{2\pi }{T}$

So, $(\frac{2\pi }{T} )^2 * r = \frac{GM}{r^2}$

We solve for T

$\frac{4\pi^2r }{T^2} = \frac{GM}{r^2}\\\\T = \sqrt{\frac{4\pi ^2r^3}{GM} }$

We know that, gravitational constant; $G = 6.67*10^{-11} m^3 /kg s^2$

We substitute in our values into the equation

$T= \sqrt{\frac{4\pi ^2(3.40*10^6 m)^3 }{(6.67*10^{-11} m^3/ kg s^2)(6.42*10^{23} kg)} }\\ \\T \sqrt{\frac{1.5516*10^{21}m^3}{4.282*10^{13}m^3/s^2} } \\\\T = \sqrt{36235404}\\\\T = 6019.585S\\\\T = 1.67hrs$

Therefore, the time interval that Mars would have to complete one rotation on its axis to make the bathroom scale have a zero reading is 1.67 Hours

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