Question

The more powerful the gravitational force of a planet, the greater its escape speed, \(v,\) and the greater the gravitational acceleration, \(g\), at its surface. However, in Table 12.1 , the value for \(v\) is much greater for Uranus than for Earth - but \(g\) is smaller on Uranus than on Earth! How can this be?

Short answer

Question: Explain why the escape speed for Uranus is greater than that for Earth, but the gravitational acceleration is smaller on Uranus than on Earth. Answer: The escape speed of Uranus is greater than that of Earth because the mass of Uranus is much greater, while its radius is also significantly larger, resulting in a higher value in the escape speed formula. However, the gravitational acceleration on Uranus is smaller than on Earth because the mass-to-radius ratio is lower for Uranus, leading to a lower value in the gravitational acceleration formula.

Step-by-step solution

Gravitational acceleration formula

The formula for gravitational acceleration (g) at the surface of a planet is given by: g = G * (M / R^2) where G is the gravitational constant (6.674 × 10^-11 N m²/kg²), M is the mass of the planet, and R is the radius of the planet.

Escape speed formula

The formula for escape speed (v) is given by: v = sqrt(2 * G * M / R) where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.

Escape speed and gravitational acceleration relationship

Notice that both formulas have a ratio of mass to radius, which is important for understanding the given observation. If a planet has a high mass and a low radius, its gravitational acceleration is high. If a planet has a high mass but a large radius, its escape speed is relatively high.

Compare Earth and Uranus

We know from the exercise that the escape speed (v) for Uranus is greater than for Earth, and the gravitational acceleration (g) is smaller on Uranus than on Earth. To understand this observation, let's compare the mass and radius of Earth and Uranus. Mass of Earth (M_E) = 5.97 × 10^24 kg Radius of Earth (R_E) = 6,371,000 m Mass of Uranus (M_U) = 86.8 × 10^24 kg Radius of Uranus (R_U) = 25,362,000 m

Analyze mass to radius ratios

Let's analyze the mass-to-radius ratios for both planets: M_E / R_E^2 = (5.97 × 10^24 kg) / (6,371,000 m)^2 = 1473.1 kg/m^2 M_U / R_U^2 = (86.8 × 10^24 kg) / (25,362,000 m)^2 = 1075.7 kg/m^2 Since the mass-to-radius ratio is higher for Earth than for Uranus (1473.1 > 1075.7), we can conclude that the gravitational acceleration (g) is greater on Earth than on Uranus.

Analyze escape speed factors

Now let's analyze the escape speed factors: sqrt(M_E / R_E) = sqrt((5.97 × 10^24 kg) / (6,371,000 m)) = 1.08 × 10^13 kg/m^3 sqrt(M_U / R_U) = sqrt((86.8 × 10^24 kg) / (25,362,000 m)) = 1.62 × 10^13 kg/m^3 Since the square root of the mass-to-radius ratio is higher for Uranus than for Earth (1.62 × 10^13 > 1.08 × 10^13), we can conclude that the escape speed (v) is greater for Uranus than for Earth. In summary, the escape speed of Uranus is greater than that of Earth because the mass of Uranus is much greater, while its radius is also significantly larger - leading to a higher value in the escape speed formula. At the same time, the gravitational acceleration on Uranus is smaller than on Earth because the mass-to-radius ratio is lower for Uranus, which results in a lower value in the gravitational acceleration formula.

Key concepts

Escape Speed

Escape speed is the minimum speed an object needs in order to break free from a planet's gravitational pull, without any additional propulsion. This speed depends on the gravitational constant, the mass of the planet, and its radius. The formula to calculate escape speed is given by:

• \(v = \sqrt{2 \cdot G \cdot \frac{M}{R}}\)

Here:

• \(v\) is the escape speed.
• \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \text{N m}^2/\text{kg}^2\).
• \(M\) is the mass of the planet.
• \(R\) is the radius of the planet.

The higher the mass of the planet, and the lower its radius, the greater the escape speed. This explains why Uranus, despite having a lower gravitational acceleration compared to Earth, has a much higher escape speed due to its immense mass.

Mass-to-Radius Ratio

The mass-to-radius ratio is a crucial factor in understanding both gravitational acceleration and escape speed. This ratio involves the mass of the planet divided by the square of its radius.
Why is this important? Let's look at the formulas:

• For gravitational acceleration: \(g = \frac{G \cdot M}{R^2}\).
• For escape speed: \(v = \sqrt{2 \cdot G \cdot \frac{M}{R}}\).

In these formulas:

• \(M/R^2\) directly influences \(g\), the gravitational acceleration, showing that a small radius with a large mass results in higher gravitational pull.
• \(M/R\) affects \(v\), the escape speed, indicating a high mass with even moderately large radius can lead to increased escape speed.

This mass-to-radius ratio helps explain the difference in values for Earth compared to Uranus, highlighting the impacts of planetary size and density on gravitational characteristics.

Gravitational Force

Gravitational force is the attractive force exerted by a planet, drawing objects towards its center. It's one of the fundamental forces in nature, and is pivotal in maintaining planetary orbits and paths. The gravitational force on the surface of a planet is influenced by the planet’s mass and radius. Using the equation:

• \(F = G \cdot \frac{M \cdot m}{R^2}\)

where:

• \(F\) is the gravitational force.
• \(m\) is the mass of the object experiencing the force.
• \(G\) is the gravitational constant.
• \(M\) and \(R\) are the mass and radius of the planet respectively.

Understanding this force aids in comprehending phenomena such as orbits, tides, and even the experience of weight on different planets. This force is also intimately connected to the concepts of escape speed and gravitational acceleration, making it a key element in planetary physics.

Planetary Physics

Planetary physics explores how the fundamental principles of physics apply to planets and other celestial bodies. It includes the study of gravitational effects, such as acceleration and force, influencing how planets interact within a solar system.
Gravitational theories and equations allow us to predict motions, structures, and evolving conditions of planets. This underpins our understanding of phenomena like planetary orbits, rotations, and atmospheres.

• Gravitational acceleration \(g\) is critical in determining surface weight and the ability of a planet to hold an atmosphere.
• Escape speed \(v\) helps ascertain whether gases or space vehicles can overcome a planet’s gravitational pull.

By studying these elements, planetary physics provides pivotal insights into how planets form, evolve, and interact, offering a greater understanding of our own planet's behavior as well as that of others in the Universe.