Homework For Lab 6 Gravitational Forces Answers

Homework for Lab 6: Gravitational Forces - Answers and In-Depth Explanations

This comprehensive guide provides detailed answers and explanations for a hypothetical Lab 6 focusing on gravitational forces. Since I don't have access to your specific lab manual or assignment, I'll cover common concepts and problem types encountered in a typical introductory physics lab on this topic. This will allow you to apply these solutions and explanations to your specific homework problems. Remember to always refer to your lab manual and lecture notes for specific instructions and formulas.

Understanding Gravitational Force

Before diving into the answers, let's review the fundamental principles governing gravitational forces. Newton's Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this is expressed as:

F = G * (m1 * m2) / r²

Where:

• F represents the gravitational force between the two objects.
• G is the gravitational constant (approximately 6.674 x 10⁻¹¹ N⋅m²/kg²).
• m1 and m2 are the masses of the two objects.
• r is the distance between the centers of the two objects.

Common Lab Exercises and Corresponding Solutions

Here, we'll explore several common lab exercises related to gravitational forces and provide detailed solutions. These examples cover a range of difficulty levels, from basic calculations to more complex analyses.

Exercise 1: Calculating Gravitational Force between Two Objects

Problem: Calculate the gravitational force between two objects with masses of 5 kg and 10 kg, separated by a distance of 2 meters.

Solution:

1. Identify the known values: m1 = 5 kg, m2 = 10 kg, r = 2 m, G = 6.674 x 10⁻¹¹ N⋅m²/kg².

2. Apply Newton's Law of Universal Gravitation:

   F = G * (m1 * m2) / r² = (6.674 x 10⁻¹¹ N⋅m²/kg²) * (5 kg * 10 kg) / (2 m)²

3. Calculate the force:

   F ≈ 8.34 x 10⁻¹⁰ N

Therefore, the gravitational force between the two objects is approximately 8.34 x 10⁻¹⁰ N. This illustrates how weak the gravitational force is at everyday scales.

Exercise 2: Analyzing the Relationship between Force and Distance

Problem: How does the gravitational force between two objects change if the distance between them is doubled? Tripled? Halved?

Solution:

This problem demonstrates the inverse square relationship in Newton's Law.

• Doubled Distance: If 'r' is doubled (becomes 2r), the force becomes F = G * (m1 * m2) / (2r)² = F / 4. The force is reduced to one-quarter of its original value.

• Tripled Distance: If 'r' is tripled (becomes 3r), the force becomes F = G * (m1 * m2) / (3r)² = F / 9. The force is reduced to one-ninth of its original value.

• Halved Distance: If 'r' is halved (becomes r/2), the force becomes F = G * (m1 * m2) / (r/2)² = 4F. The force is increased to four times its original value.

Exercise 3: Determining the Mass of an Object Using Gravitational Force

Problem: Two objects are placed at a distance of 1 meter apart. The known object has a mass of 20 kg. The gravitational force measured between them is 1 x 10⁻⁸ N. Calculate the mass of the unknown object.

Solution:

1. Rearrange the formula to solve for the unknown mass (m2):

   m2 = (F * r²) / (G * m1)

2. Substitute the known values:

   m2 = (1 x 10⁻⁸ N * (1 m)²) / ((6.674 x 10⁻¹¹ N⋅m²/kg²) * 20 kg)

3. Calculate the mass:

   m2 ≈ 7.5 kg

Therefore, the mass of the unknown object is approximately 7.5 kg.

Exercise 4: Analyzing Gravitational Field Strength

Problem: Calculate the gravitational field strength (g) at a distance of 6,371 km (Earth's radius) from the center of the Earth. The mass of the Earth is approximately 5.972 x 10²⁴ kg.

Solution:

Gravitational field strength is the force per unit mass experienced by an object at a given point in a gravitational field. It's calculated as:

g = G * M / r²

Where:

• g is the gravitational field strength.
• G is the gravitational constant.
• M is the mass of the Earth.
• r is the distance from the center of the Earth.

1. Convert the distance to meters: r = 6,371,000 m

2. Substitute the known values:

   g = (6.674 x 10⁻¹¹ N⋅m²/kg²) * (5.972 x 10²⁴ kg) / (6,371,000 m)²

3. Calculate the gravitational field strength:

   g ≈ 9.81 m/s²

Therefore, the gravitational field strength at the Earth's surface is approximately 9.81 m/s², which is consistent with the standard acceleration due to gravity.

Exercise 5: Advanced Problem – Orbital Mechanics (Kepler's Laws)

Problem: A satellite is orbiting Earth at a specific altitude. Given the satellite's orbital period, determine its orbital radius. (This problem would require using Kepler's Third Law and assuming a circular orbit).

Solution: Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. For a circular orbit, the semi-major axis is the orbital radius (r). Mathematically:

T² ∝ r³ or, with a constant of proportionality: T² = k * r³

Where:

• T is the orbital period.
• r is the orbital radius.
• k is a constant that depends on the mass of the central body (Earth in this case). For orbits around Earth, k = 4π²/GM_Earth.

To solve this problem you would need to:

1. Determine the constant k: Using the gravitational constant (G) and the mass of the Earth (M_Earth).
2. Substitute the known orbital period (T) into Kepler's Third Law: Solve for r³.
3. Calculate the orbital radius (r): Find the cube root of the result.

These examples provide a foundation for understanding and solving various homework problems related to gravitational forces. Remember to always:

• Clearly define the known variables.
• Choose the appropriate formula.
• Pay close attention to units.
• Double-check your calculations.

This detailed guide, though hypothetical in its lab exercises, gives a comprehensive look at the types of problems you might encounter, allowing you to better tackle your actual Lab 6 assignment. Remember to always consult your specific lab instructions and textbook for the most accurate information relevant to your coursework.