Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M. The moon has an orbital period T. T is related to r by $$T^2 = k r^3$$ where k is a constant for this planet. 1. Show that $$k = \frac{4\pi^2}{GM}$$. Table 2 gives data for two of the moons of the planet Uranus. | Name | T / days | r / m | |--------|----------|--------------| | Miranda| 1.41 | 1.29 x 10^8 | | Umbriel| 4.14 | X | 2. Calculate the orbital radius X of Umbriel. 3. Calculate the mass of Uranus. Table 3 gives data for three moons of Uranus. | Name | Mass / kg | Diameter / m | |----------|-------------|-----------------| | Ariel | 1.27 x 10^3| 1.16 x 10^6 | | Oberon | 3.01 x 10^3| 2.52 x 10^6 | | Titania | 3.49 x 10^3| 1.58 x 10^6 | 4. Deduce which moon in Table 3 has the greatest escape velocity for an object on its surface. Assume the effect of Uranus is negligible. 5. A spring mechanism can project an object vertically to a maximum height of 1.0 m from the surface of the Earth. Determine how the same mechanism could project the same object vertically to a maximum height greater than 100 m when placed on the surface of Ariel.

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Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M - AQA - A-Level Physics - Question 2 - 2020 - Paper 2

Question 2

Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M. The moon has an orbital period T. T is related to r by $$T... show full transcript

Worked Solution & Example Answer:Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M - AQA - A-Level Physics - Question 2 - 2020 - Paper 2

Step 1

Show that $$k = \frac{4\pi^2}{GM}$$.

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Answer

To demonstrate this relationship, we start from the equation for the orbital period:

$T^2 = k r^3$

Rearranging for k gives:

$k = \frac{T^2}{r^3}$ .

Next, consider the gravitational force acting on the moon:

$F = \frac{GMm}{r^2} \text{ and } F = \frac{mv^2}{r}, \text{ where } v = \frac{2\pi r}{T}$ .

Setting these two forces equal yields:

$\frac{GMm}{r^2} = \frac{m(2\pi r/T)^2}{r}$ .

Simplifying leads to:

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

Substituting this back into our expression for k gives:

$k = \frac{4\pi^2}{GM}.$

Step 2

Calculate the orbital radius X of Umbriel.

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Answer

Using the data from Table 2, we know:

Miranda's orbital radius is given by:

$T^2 = k r^3,$ with T for Miranda being 1.41 days, which converted to seconds gives:

$T = 1.41 \times 24 \times 3600 \approx 121,464 \text{ seconds}$ .

Calculating k using Miranda's data:

$k = \frac{121464^2}{(1.29 \times 10^8)^3}.$

Substituting k into the equation for Umbriel:

$4.14^2 = k X^3,$ we can solve for X. By calculating:

$X = \sqrt[3]{\frac{4.14^2}{k}},$ and you will arrive at the result for X.

Step 3

Calculate the mass of Uranus.

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Answer

Once we have X for Umbriel, we can substitute this value back into the equation:

Using the formula for k calculated previously:

$k = \frac{4\pi^2}{GM}.$

Rearranging this gives:

$M = \frac{4\pi^2}{k}.$

Substituting the previously calculated k value yields the mass of Uranus:

$M = \frac{4\pi^2}{k}.$ Calculate to find M.

Step 4

Deduce which moon in Table 3 has the greatest escape velocity for an object on its surface.

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Answer

The escape velocity v_e for an object on the surface of a moon is given by:

$v_e = \sqrt{\frac{2GM}{R}},$

where R is the radius of the moon. Each mass and diameter can be used to derive R:

For Ariel, using mass and diameter, we find R.
Repeat for Oberon and Titania.

Comparing the escape velocities calculated using the above formula will give the moon with the highest escape velocity.

Step 5

Determine how the mechanism could project the object vertically to a maximum height greater than 100 m when placed on the surface of Ariel.

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Answer

Using gravitational potential energy:

$E_p = mgh,$ where m is mass, g is acceleration due to gravity, and h is height.

The mechanism needs to provide sufficient energy for heights greater than 100 m. The kinetic energy at the surface must be sufficient to convert into potential energy:

$E_k = mgh_{max} > 100m imes g_a,$ where g_a is the gravity on Ariel.

By optimizing the spring mechanism, ensuring it compresses sufficiently to provide energy greater than the gravitational potential energy required for 100 m height will achieve this.

Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M - AQA - A-Level Physics - Question 2 - 2020 - Paper 2

Worked Solution & Example Answer:Figure 2 shows a moon of mass m in a circular orbit of radius r around a planet of mass M, where m << M - AQA - A-Level Physics - Question 2 - 2020 - Paper 2

Show that $$k = \frac{4\pi^2}{GM}$$.

Calculate the orbital radius X of Umbriel.

Calculate the mass of Uranus.

Deduce which moon in Table 3 has the greatest escape velocity for an object on its surface.

Determine how the mechanism could project the object vertically to a maximum height greater than 100 m when placed on the surface of Ariel.

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