Problem 36 - University Physics with Modern Physics

Question

You have just landed on Planet \(X\) . You take out a 100 -g ball, release it from rest from a height of \(10.0 \mathrm{m},\) and measure that it takes 2.2 \(\mathrm{s}\) to reach the ground. You can ignore any force on the ball from the atmosphere of the planet. How much does the \(100-\mathrm{g}\) ball weigh on the surface of Planet \(\mathrm{X}\) ?

Step-by-Step Solution

Answer

The 100-g ball weighs approximately 0.413 N on Planet X.

Step 1: Understand the Problem

We need to determine the weight of a 100-g (or 0.1-kg) ball on Planet X. The weight is the force of gravity acting on the ball, defined as the mass times the gravitational acceleration on Planet X.

Step 2: Use the Formula for Weight

The formula to calculate weight is given by: \[ W = m \times g \]where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the gravitational acceleration on Planet X, which we need to find.

Step 3: Use the Formula for Free-fall Motion

Since the ball is released from rest, we will use the formula for the distance covered in free fall: \[ d = \frac{1}{2}gt^2 \]where \( d \) is the distance, \( g \) is the gravitational acceleration, and \( t \) is the time taken to fall.

Step 4: Plug in the Values

In our problem, the ball falls a distance of \( 10.0 \ \text{m} \) in \( 2.2 \ \text{s} \). Substitute these values into the equation from Step 3:\[ 10.0 = \frac{1}{2}g (2.2)^2 \]

Step 5: Solve for Gravitational Acceleration

Rearrange the equation to solve for \( g \):\[ 10.0 = \frac{1}{2}g \times 4.84 \]\[ 10.0 = 2.42g \]\[ g = \frac{10.0}{2.42} \approx 4.13 \ \text{m/s}^2 \]

Step 6: Calculate the Weight of the Ball

Now use the value of \( g \) to calculate the weight \( W \):\[ W = 0.1 \times 4.13 = 0.413 \ \text{N} \]

Key Concepts

Free-fall Motion

When an object is in free-fall, it means that it is moving under the influence of gravity alone, without any resistance from air or any other force. Free-fall motion is a classic way to study gravity, as it provides a direct opportunity to observe how gravity accelerates an object.
For Planet X, we launched a ball from rest at a height of 10 meters. Free-fall equations help us determine how fast the ball falls and how gravity affects it. The fundamental equation is \[ d =\frac{1}{2} \, g \, t^2 \]where:

• \( d \) is the distance the ball travels,
• \( g \) is gravitational acceleration, and
• \( t \) is the time it takes to reach the ground.

In our example, gravity pulls the ball towards the ground in 2.2 seconds, covering a distance of 10 meters. This motion confirms the constant acceleration revealed by gravity on this planet, as the speed increases while falling. Thus, measuring free-fall motion is crucial for understanding the strength of gravity on different planets.

Weight Calculation

Weight is simply the force exerted on an object due to gravity. On any celestial body, whether it's Earth or Planet X, weight will differ depending on the local gravitational pull.
To find the weight of our 100-g ball on Planet X, we use the formula\[ W = m \times g \]where:

• \( W \) is the weight,
• \( m \) is the mass of the object (0.1 kg for our ball), and
• \( g \) is the gravitational acceleration.

We already calculated \( g = 4.13 \, \text{m/s}^2 \) based on the free-fall equation. Plugging in these values, we find \( W = 0.1 \times 4.13 = 0.413 \, \text{N} \).

This calculation shows how the weight changes with different gravitational forces even when the object's mass stays the same. Such exercises help clarify weight's dependence on gravity, exemplifying its difference from mass, which is constant irrespective of location.

Planetary Gravity

Gravity varies from planet to planet, influencing how everything from a small ball to massive objects behaves. The concept of planetary gravity is understood by how it determines the local gravity field's strength, which is the constant \( g \) used in the weight and motion equations.
On Planet X, we determined that its gravitational acceleration \( g = 4.13 \, \text{m/s}^2 \), which is different from Earth's \( 9.81 \, \text{m/s}^2 \). This difference directly influences how fast objects will fall and how heavy they will feel on its surface.

Understanding planetary gravity helps:

• Predict motion of objects.
• Calculate weight of astronauts and equipment, essential for space travel.
• Explore gravitational effects on atmosphere and environment.

By analyzing free-fall data, scientists and engineers can design structures and tools tailored specifically for that planet’s unique pull, ensuring safe and functional applications in exploratory missions.