Problem 19 - University Physics with Modern Physics

Question

At the surface of Jupiter's moon Io, the acceleration due to gravity is g = 1.81 m/s\(^2\). A watermelon weighs 44.0 N at the surface of the earth. (a) What is the watermelon's mass on the earth's surface? (b) What would be its mass and weight on the surface of Io?

Step-by-Step Solution

Answer

Mass on Earth: 4.49 kg; mass on Io: 4.49 kg; weight on Io: 8.12 N.

Step 1: Find the mass of the watermelon on Earth's surface

To find the mass of the watermelon on Earth's surface, use the formula for weight: \( W = m \cdot g \). Here, \( W = 44.0 \) N is the weight of the watermelon, and \( g = 9.8 \) m/s\(^2\) is the acceleration due to gravity on Earth. Rearrange the formula to find mass: \( m = \frac{W}{g} \). Substitute the given values: \( m = \frac{44.0}{9.8} \approx 4.49 \) kg.

Step 2: Determine the mass of the watermelon on Io's surface

The mass of an object remains constant regardless of its location. Therefore, the mass of the watermelon on Io's surface is the same as on Earth's surface, which is 4.49 kg.

Step 3: Calculate the weight of the watermelon on Io's surface

Weight can be calculated using the formula: \( W = m \cdot g_{\text{Io}} \), where \( m = 4.49 \) kg and \( g_{\text{Io}} = 1.81 \) m/s\(^2\) is Io's gravitational acceleration. Substitute these values to find the weight: \( W = 4.49 \times 1.81 \approx 8.12 \) N.

Key Concepts

Mass

Mass is an intrinsic property of an object that signifies the amount of matter it contains. It's important to note that mass remains constant regardless of an object’s location.
On Earth or on the moon Io, the mass of an object does not change because it does not depend on gravity.

• Mass is measured in kilograms (kg) in the International System of Units (SI).
• The concept of mass is often confused with weight, but they are fundamentally different. While weight depends on gravity, mass does not.
• In the exercise, we calculated the mass of a watermelon which was 4.49 kg, using its weight on Earth and the gravitational force there.

Understanding mass is crucial when determining how an object relates to gravity, as it helps in the calculation of weight.

Weight

Weight is the force exerted on an object due to gravity. It is determined by the mass of the object and the acceleration due to gravity.
On different planets or moons, where gravity varies, an object's weight will change even though its mass remains the same.

• Weight is measured in newtons (N), a unit derived from kilograms and meters per second squared.
• The formula to calculate weight is: \[ W = m \cdot g \]where \( m \) is mass and \( g \) is the gravitational acceleration.
• In Io's case, the watermelon’s weight was calculated to be 8.12 N using its constant mass and Io’s reduced gravity.

Thus, weight is essentially how heavy something feels at a location with a specific gravitational pull. It's fascinating to see how weight can change just by relocating an object in space.

Acceleration Due to Gravity

Acceleration due to gravity is the rate at which an object accelerates when it is falling solely under the influence of gravity. On different celestial bodies, this acceleration can vary significantly.
For Earth, this value is about 9.8 m/s², but on Io, it is much less at 1.81 m/s².

• This means that an object will fall much slower on Io than on Earth.
• The acceleration due to gravity directly affects weight but does not affect mass.
• It plays a pivotal role in determining how quickly an object will speed up as it falls.

In summary, understanding acceleration due to gravity is key in natural sciences, especially when studying the differences between celestial bodies. It explains why the same object weighs less on Io compared to Earth, as seen through the watermelon example.