Problem 12 - University Physics with Modern Physics

Question

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venus?

Step-by-Step Solution

Answer

(a) 8.87 m/s², (b) 67.9 N

Step 1: Understanding the Formula for Gravity

To compute the acceleration due to gravity on Venus, we need to use the formula for gravitational acceleration, given by \( g = \frac{GM}{R^2} \), where \( G \) is the gravitational constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet. We will use Earth's gravitational acceleration \( g_e = 9.81 \, \text{m/s}^2 \) to simplify our calculations.

Step 2: Applying Mass and Radius Ratios

The mass of Venus is 81.5% that of Earth, so \( M_v = 0.815M_e \). The radius of Venus is 94.9% that of Earth, so \( R_v = 0.949R_e \). By substituting these values into the formula for gravity and considering the ratio \( g_v = \frac{G M_v}{R_v^2} \), we can express \( g_v \) in terms of \( g_e = \frac{G M_e}{R_e^2} \).

Step 3: Simplifying the Expression

From the expressions for \( g_v \) and \( g_e \), we find \( g_v = g_e \times \frac{0.815}{0.949^2} \). By computing this, we find that the acceleration due to gravity on the surface of Venus is \( g_v \approx 8.87 \, \text{m/s}^2 \).

Step 4: Calculating Rock's Weight on Venus

Weight is the force experienced by a mass due to gravity, given by \( W = mg \). On Earth, the weight \( W_e = mg_e = 75.0 \, \text{N} \). The mass \( m \) of the rock can be calculated as \( m = \frac{75.0}{g_e} \). Compute \( m \) and use it to find the weight on Venus: \( W_v = m g_v \).

Step 5: Final Weight Calculation

Calculate the mass: \( m = \frac{75.0}{9.81} \approx 7.65 \, \text{kg} \). Then compute the weight on Venus: \( W_v = m g_v = 7.65 \times 8.87 \approx 67.9 \, \text{N} \). Thus, the rock weighs approximately 67.9 N on the surface of Venus.

Key Concepts

Acceleration Due to Gravity

The acceleration due to gravity is a measure of how quickly an object will accelerate when it is in free fall, ignoring air resistance, toward the surface of a celestial body like a planet.
This acceleration is denoted by the symbol \( g \) and can be calculated using the formula \( g = \frac{GM}{R^2} \). Here, \( G \) represents the gravitational constant, a universal constant, \( M \) is the mass of the planet, and \( R \) is the radius of the planet.
The concept is crucial to understanding how gravity varies across different planets. For Earth, this value is typically \( g = 9.81 \, \text{m/s}^2 \).

• When calculating on other planets, it's important to adjust this formula based on the specific mass and radius values of that planet.
• Different planets will have different gravitational accelerations due to their unique masses and radii.
• Understanding the acceleration due to gravity helps in calculating how fast an object will fall and how heavy it will feel on the surface of different planets.

Mass and Radius Ratios

The mass and radius of a planet significantly determine its gravitational force, which in turn affects the acceleration due to gravity. When comparing the planets, like Earth and Venus, these two factors need careful consideration.
For instance, Venus has 81.5% the mass of Earth and 94.9% of Earth's radius. These percentage values are used as ratios when calculating gravity on Venus compared to Earth.

• These mass and radius ratios allow us to express Venus's gravitational acceleration, \( g_v \), in terms of Earth's gravitational acceleration, \( g_e \).
• By substituting the ratios into the formula, we analyze how Venus's lesser mass and slightly smaller radius influence gravity compared to Earth.
• This comparison is essential for understanding the differences in physical experiences, like weight, between these two planets.

The formula adjustment based on Venus's mass and radius ratio is a critical step in ensuring that calculations are accurate.

Weight Calculation on Different Planets

Weight is the force exerted by gravity on an object's mass. Interestingly, an object will weigh differently on other planets due to changes in gravitational acceleration.
To determine how much an object, such as a rock, weighs on Venus, we must first know its weight on Earth, calculated by \( W_e = m \times g_e \), where \( m \) is the mass.

• The weight on Venus, \( W_v \), can be found using the formula \( W_v = m \times g_v \).
• First, we calculate the mass \( m \) of the rock on Earth using \( m = \frac{W_e}{g_e} \).
• Then, we apply this mass in \( W_v \) to find the weight on Venus using Venus's gravity \( g_v \).

For instance, a rock weighing 75 N on Earth will weigh approximately 67.9 N on Venus. This example highlights how gravity's variation affects weight, helping us understand experiences of weightlessness or increased weight on various planets.