Problem 78 - University Physics with Modern Physics

Question

The planet Uranus has a radius of \(25,560 \mathrm{km}\) and a surface acceleration due to gravity of 11.1 \(\mathrm{m} / \mathrm{s}^{2}\) at its poles. Its moon Miranda (discovered by Kuiper in 1948 is in a circular orbit about Uranus at an altitude of \(104,000 \mathrm{km}\) above the planet's surface. Miranda has a mass of \(6.6 \times 10^{19} \mathrm{kg}\) and a radius of 235 \(\mathrm{km}\). (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Miranda's acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Miranda's gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released 1 \(\mathrm{m}\) above Miranda's surface on the side toward Uranus will fall \(u p\) relative to Miranda? Explain.

Step-by-Step Solution

Answer

(a) Mass of Uranus \( \approx 8.68 \times 10^{25} \) kg. (b) Miranda's orbital acceleration \( \approx 0.0145 \) m/s². (c) Gravity on Miranda \( \approx 0.079 \) m/s². (d) The object falls towards Miranda, not up.

Step 1: Calculate Mass of Uranus

To find the mass of Uranus, use the formula for gravitational acceleration \( g = \frac{G M_U}{R_U^2} \), where \( g = 11.1 \, \text{m/s}^2 \), \( R_U = 25,560,000 \, \text{m} \), and \( G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \). Rearrange to find Uranus's mass:\[ M_U = \frac{g R_U^2}{G} = \frac{11.1 \times (25,560,000)^2}{6.674 \times 10^{-11}} \approx 8.68 \times 10^{25} \, \text{kg} \]

Step 2: Calculate Miranda's Orbital Acceleration

Miranda's altitude above Uranus is \( 104,000 \, \text{km} \). Add this to Uranus's radius for total distance from Uranus's center: \( 25,560 + 104,000 = 129,560 \, \text{km} = 129,560,000 \, \text{m} \). Use the formula for centripetal acceleration \( a = \frac{G M_U}{r^2} \):\[ a = \frac{6.674 \times 10^{-11} \times 8.68 \times 10^{25}}{(129,560,000)^2} \approx 0.0145 \, \text{m/s}^2 \]

Step 3: Calculate Gravity on Miranda's Surface

For the gravitational acceleration on Miranda's surface use \( g_m = \frac{G M_m}{R_m^2} \), where \( M_m = 6.6 \times 10^{19} \, \text{kg} \) and \( R_m = 235,000 \, \text{m} \):\[ g_m = \frac{6.674 \times 10^{-11} \times 6.6 \times 10^{19}}{(235,000)^2} \approx 0.079 \, \text{m/s}^2 \]

Step 4: Determine Motion of Object Released Near Miranda

Compare the gravitational acceleration of Miranda \(0.079 \, \text{m/s}^2\) with the centripetal acceleration due to Uranus \(0.0145 \, \text{m/s}^2\). Since the gravitational pull by Miranda is stronger, an object released on the side toward Uranus will "fall down" towards Miranda rather than "falling up" due to Uranus's weaker pull at that location.

Key Concepts

Mass of Uranus

Understanding the mass of a planet like Uranus requires understanding the synergy between gravitational forces and physical dimensions. When you look at a planet, the gravitational acceleration it exhibits is a direct consequence of its mass and radius. The mass of Uranus can be calculated using the formula for gravitational acceleration:

• Gravitational acceleration ( \( g \)): The acceleration that a body experiences due to the gravitational pull of another massive body. For Uranus, it is given as 11.1 \( \mathrm{m} / \mathrm{s}^2 \).
• Gravitational constant ( \( G \)): A fundamental constant \( 6.674 \times 10^{-11} \mathrm{N} \mathrm{m}^2/\mathrm{kg}^2 \) used in the calculation.
• Radius of Uranus ( \( R_U \)): Provided as 25,560 km, or 25,560,000 meters.

We rearrange the formula: \( g = \frac{G M_U}{R_U^2} \) to solve for the mass of Uranus (\( M_U \)). This provides the enormous mass: \( \approx 8.68 \times 10^{25} \mathrm{kg} \). Using this mass provides insights into the immense gravitational pull Uranus exerts on surrounding bodies, such as its moons.

Orbital Motion

Orbital motion describes how a moon like Miranda moves around Uranus. This motion is influenced by the gravitational force that Uranus exerts on Miranda, keeping it in orbit. Key aspects include:

• Circular Orbit: Miranda is in a near circular orbit around Uranus, which simplifies calculations.
• Altitude: Miranda orbits 104,000 km above Uranus's surface, meaning the total distance from Uranus’s center to Miranda’s center of orbit is 129,560 km (by adding Uranus's radius).

With a known distance, we can apply the formula \( a = \frac{G M_U}{r^2} \) to find Miranda's centripetal acceleration.

• Centripetal Acceleration: This acceleration is directed towards the center of Uranus, allowing Miranda to maintain its stable orbit. It's calculated to be \( \approx 0.0145 \mathrm{m/s}^2 \).

Understanding Miranda's orbital path allows us to gauge the precise nature of its circular journey around Uranus.

Centripetal Acceleration

Centripetal acceleration is key to understanding how objects remain in orbit. It is the inward force needed to keep an object moving in a circular path.- **Formula:** Centripetal acceleration ( \( a \) ) is given by \( a = \frac{v^2}{r} \), or more usefully in gravitational contexts as \( a = \frac{G M}{r^2} \).- **Balancing Forces:** The force of gravity provides the centripetal force necessary for this circular motion.For an object like Miranda, the gravitational attraction from Uranus supplies the necessary centripetal acceleration to keep it bound in its orbit. While this acceleration is small (\( 0.0145 \mathrm{m/s}^2 \)), it illustrates how even minimal forces can maintain moons and satellites in their paths around massive planets over long periods.

Gravitational Force Calculation

Calculating gravitational force involves understanding how two bodies, such as Uranus and its moon Miranda, exert force on each other. The force can be calculated through:

• **Newton’s Law of Universal Gravitation:** Describes the gravitational force as \( F = \frac{G M_1 M_2}{r^2} \).
• **Key Elements:**
  • \( M_1 \) and \( M_2 \) represent the masses of Uranus and Miranda, respectively.
  • The distance \( r \) is the total distance between the centers of the planets (Uranus’s radius plus Miranda’s orbit altitude).

This explains how the massive gravitational pull of Uranus can influence Miranda and maintain its orbit. Furthermore, comparing gravitational forces like those between Miranda and Uranus can illustrate how different forces act when interacting with smaller distances like Miranda’s surface gravity and the effects of centripetal forces in massive celestial mechanics.