Dynamics of Rigid Bodies: The Concept of Law of Gravitation

  • #1
Nova_Chr0n0
16
3
Homework Statement
Find the mass of an object (in kg) that weighs 2 kN at a height of 1800 km above the earth’s surface.
Relevant Equations
Mass of earth - 5.9742 x 10^24 kg
Radius of earth = 6378 km
F=G[(m_a*m_b)/R^2]
The questions and relevant formulas/information are attached below:

1701242900601.png

1701242922499.png

1701243032043.png


I started by writing the needed values:

2 kN = 2,000 N
6378 km =6378(1000) m
1800 km = 1800(1000)m

1701244278069.png


After converting, I tried solving for the mass using the law of gravitation formula:

1701244303798.png


I decided to replace N as (kgm)/s^2:

1701244326240.png

Solving for the mass:

1701244347866.png


I wanted to verify my final answer by trying to find it in the internet. I only saw 1 solution and we got the same values up until mass B. For his mass B, he got 335.4695 N but I got 335.721 kg. The other solution will be shown below:

1701244878834.png
1701244891704.png


Am I suppose to get mass B here with N as its unit? If so, on what part did I do the algebra wrong?
 
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  • #2
Mass is measured in kg. Your numerical answer looks correct, although you may think about learning some algebra!
 
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FAQ: Dynamics of Rigid Bodies: The Concept of Law of Gravitation

What is the Law of Gravitation?

The Law of Gravitation, formulated by Sir Isaac Newton, states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is mathematically represented as F = G * (m1 * m2) / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance between their centers.

How does the Law of Gravitation apply to rigid bodies?

In the context of rigid bodies, the Law of Gravitation is used to determine the gravitational forces acting between objects that do not deform under the influence of these forces. The gravitational force on a rigid body can be calculated by considering the body as a collection of point masses, integrating the contributions of these point masses, or by assuming the mass is concentrated at the center of mass for simpler calculations.

What is the significance of the gravitational constant (G) in the Law of Gravitation?

The gravitational constant (G) is a proportionality factor that appears in the equation for Newton's Law of Gravitation. Its value is approximately 6.674 × 10^-11 N m²/kg². G is crucial because it quantifies the strength of the gravitational force and allows us to calculate the force of attraction between two masses. Without G, the relationship between mass, distance, and gravitational force could not be accurately determined.

How do you calculate the gravitational force between two rigid bodies?

To calculate the gravitational force between two rigid bodies, you can use the formula F = G * (m1 * m2) / r^2. Here, m1 and m2 are the masses of the two bodies, r is the distance between their centers of mass, and G is the gravitational constant. For complex shapes, you may need to integrate the gravitational forces over the volume of the bodies, but for many practical purposes, treating the masses as point masses located at their centers of mass is sufficient.

What role does the center of mass play in the dynamics of rigid bodies under gravitational forces?

The center of mass is the point at which the entire mass of a body can be considered to be concentrated for the purpose of analyzing translational motion under external forces, including gravitational forces. In the dynamics of rigid bodies, the gravitational force can be assumed to act at the center of mass, simplifying the analysis of the motion of the body under gravity. This allows us to treat complex bodies as if they were point masses located at their centers of mass when calculating gravitational interactions.

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