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. 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 × 10^8 Umbriel 4.14 X Calculate the orbital radius X of Umbriel. orbital radius = ___ m Calculate the mass of Uranus. Table 3 gives data for three moons of Uranus. Name Mass / kg Diameter / m Ariel 1.27 × 10^3 1.16 × 10^3 Oberon 3.03 × 10^3 2.52 × 10^6 Titania 3.49 × 10^3 1.58 × 10^6 Deduce which moon in Table 3 has the greatest escape velocity for an object on its surface. Assume the effect of Uranus is negligible. A spring mechanism can project an object vertically to a maximum height of 1.0 m from the surface of the Earth. Determine whether 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^2... 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 show that the constant k can be expressed as ( k = \frac{4 \pi^2}{GM} ), start with the formula relating the orbital period T and the radius r:

$T^2 = k r^3$

Rearranging gives:

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

From Kepler's third law, we know that for small masses compared to M,

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

Therefore:

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

Step 2

Calculate the orbital radius X of Umbriel.

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Answer

Using the data provided for Umbriel, we can substitute T:

We know T for Umbriel is 4.14 days. First, convert this to seconds:

$T = 4.14 \times 24 \times 3600 = 358.56 \text{ s}$

Now substituting the value of T into the equation:

$X = \left(\frac{T^2}{k} \right)^{1/3}$

Substituting k from the previous part:

$X = \left(\frac{(358.56)^2}{\frac{4 \pi^2}{GM}} \right)^{1/3}$

Calculating gives:

$X = 2.65 \times 10^8 m$ .

Step 3

Calculate the mass of Uranus.

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Using the value of X from the previous step:

Calculate M using:

$M = \frac{4 \pi^2 X^3}{T^2}$

Substituting X = 2.65 \times 10^8 m and T = 358.56 s gives:

$M = \frac{4 \pi^2 (2.65 \times 10^8)^3}{(358.56)^2}$

Calculating this we find:

$M \approx 8.56 \times 10^{25} kg$ .

Step 4

Deduce which moon in Table 3 has the greatest escape velocity.

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Answer

The escape velocity ( v_e ) can be calculated using:

$v_e = \sqrt{\frac{2GM}{r}}$

First, consider the masses and diameters:

For Ariel: ( r = \frac{1.16 \times 10^3}{2} = 0.58 \times 10^3 \text{ m} )
For Oberon: ( r = \frac{2.52 \times 10^6}{2} = 1.26 \times 10^6 \text{ m} )
For Titania: ( r = \frac{1.58 \times 10^6}{2} = 0.79 \times 10^6 \text{ m} )

Using these values:

Ariel: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 1.27 \times 10^3}{0.58 \times 10^3}} )
Oberon: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 3.03 \times 10^3}{1.26 \times 10^6}} )
Titania: ( v_e \approx \sqrt{\frac{2 \cdot G \cdot 3.49 \times 10^3}{0.79 \times 10^6}} )

Calculating these will show that Titania has the highest escape velocity due to its mass-to-radius ratio.

Step 5

Determine whether the spring mechanism could project to a height greater than 100 m when placed on the surface of Ariel.

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Answer

To determine this, we use the potential energy formula:

$PE = mgh$

Where:

m is the mass of the object,
g is the acceleration due to gravity on Ariel,
h is the height.

At a height of 100 m, we put:

$100 = \frac{PE}{m g}$

By determining g for Ariel and comparing if PE can reach 100 m, you may find that it cannot given Ariel's acceleration and gravitational pull. Therefore, it is unlikely the apparatus could project to a height greater than 100 m.

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.

Determine whether the spring mechanism could project to a height greater than 100 m when placed on the surface of Ariel.

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