Problem 21 - College Physics

Question

A scientific instrument that weighs 85.2 \(\mathrm{N}\) on the earth weighs 32.2 \(\mathrm{N}\) at the surface of Mercury. (a) What is the acceleration due to gravity on Mercury? (b) What is the instrument's mass on earth and on Mercury?

Step-by-Step Solution

Answer

a) Gravity on Mercury is 3.71 m/s². b) Mass is 8.68 kg on both planets.

Step 1: Understanding the Basic Weight Equation

The weight of an object can be calculated using the formula: \[ W = m \cdot g \] where \(W\) is the weight, \(m\) is the mass, and \(g\) is the acceleration due to gravity. We will apply this formula to both Earth and Mercury.

Step 2: Calculate Mass on Earth

For the object on Earth:- Weight \(W = 85.2\,\mathrm{N}\)- Acceleration due to gravity \(g = 9.81\,\mathrm{m/s^2}\)Rearrange the weight formula to solve for \(m\): \[ m = \frac{W}{g} = \frac{85.2}{9.81} \approx 8.68\,\mathrm{kg} \] This is the mass of the object on Earth.

Step 3: Calculate Acceleration due to Gravity on Mercury

For the object on Mercury:- Weight \(W = 32.2\,\mathrm{N}\)- Mass \(m = 8.68\,\mathrm{kg}\) (since mass remains constant in different locations)Solve for \(g\) on Mercury using the weight equation: \[ g = \frac{W}{m} = \frac{32.2}{8.68} \approx 3.71\,\mathrm{m/s^2} \]Thus, the acceleration due to gravity on Mercury is approximately \(3.71\,\mathrm{m/s^2}\).

Step 4: Determine Mass on Mercury

The mass of the instrument remains the same regardless of its location. Therefore, the mass of the instrument on Mercury is the same as its mass on Earth, which is \(8.68\,\mathrm{kg}\).

Key Concepts

Weight Calculation

Calculating weight is essential in understanding how much force is exerted by gravity on an object. The weight of an object is determined by the formula:

• Weight Formula: \( W = m \cdot g \)

Where \( W \) stands for weight, \( m \) represents mass, and \( g \) is the acceleration due to gravity.
The given exercise shows how different gravitational forces affect the weight of an object. On Earth, with a stronger gravitational pull, the instrument's weight is higher. On Mercury, the gravity is weaker, resulting in a lower weight.
This directly illustrates the dependence of weight on local gravitational conditions.

Mass Conservation

Mass conservation is a fundamental concept in physics. Despite changes in location and gravitational acceleration, the mass of an object remains unchanged.
For instance, the instrument in the exercise maintains its mass of approximately \(8.68 \, \mathrm{kg}\) whether on Earth or Mercury. Understanding that mass is a measure of matter within an object helps clarify why its value doesn't fluctuate despite external changes. This consistency is crucial when solving physics problems that involve different environments.
Knowing that mass is constant allows scientists to use it as a reliable point for further calculations and analyses.

Physics Problem Solving

Physics problem solving involves applying theoretical concepts to find practical solutions. It requires a systematic approach to break down the problem into manageable steps.

• First, identify what is known—here, the weights on Earth and Mercury.
• Next, utilize relevant formulas to work through each segment, such as finding mass and gravitational acceleration.
• Finally, ensure logical consistency and check calculations to verify results.

This exercise showcases problem-solving by using equations for weight and acceleration to determine mass and gravitational forces. Such methods strengthen understanding and build confidence in tackling different physics challenges.

Gravitational Acceleration Comparison

Comparing gravitational acceleration helps highlight how gravity varies across different celestial bodies. For Earth, the gravitational acceleration is approximately \( 9.81 \, \mathrm{m/s^2} \), whereas, for Mercury, it's much lower, around \( 3.71 \, \mathrm{m/s^2} \).
This difference affects how objects behave under gravitational forces.

• Stronger gravity, like Earth's, results in higher weights for the same mass.
• Weaker gravity, such as on Mercury, means lower weights.

This concept is critical in understanding planetary environments and engineering applications across varying gravitational fields. Recognizing these differences allows scientists to prepare and adapt materials and instruments accordingly.