Predict/Calculate A spaceship of mass m travels from the...

Predict/Calculate A spaceship of mass $m$ travels from the Earth to the Moon along a line that passes through the center of the Earth and the center of the Moon. (a) At what distance from the center of the Earth is the force due to the Earth twice the magnitude of the force due to the Moon? (b) How does your answer to part (a) depend on the mass of the spaceship? Explain.

Key Concepts

Newton's Law of Universal Gravitation

This law states that every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is fundamental to calculating gravitational interactions, as it defines how the gravitational force varies with distance and mass, and is directly applied when comparing the gravitational pulls from two different bodies.

Inverse Square Law

The inverse square law is a principle that describes how a specified physical quantity (like gravitational force) decreases in proportion to the square of the distance from the source. In the context of gravitational attraction, this means that as you move further away from a massive object, the gravitational pull diminishes rapidly, a concept that is central in determining points along a line between two masses where forces have a specific ratio.

Superposition Principle of Gravitational Forces

The superposition principle allows for the calculation of the net gravitational force acting on an object by summing the individual gravitational forces exerted by multiple bodies. This concept is crucial for problems where a test mass is influenced by more than one gravitational source, as it permits the analysis of combined effects to find equilibrium or specific force ratios.

Mass Independence of Gravitational Acceleration

In gravitational systems, the acceleration due to gravity is independent of the mass of the object experiencing the force. Although the gravitational force is proportional to the mass of the object, when calculating acceleration (force divided by mass), the object’s mass cancels out. This principle explains why the relative positions where two gravitational influences have a given ratio do not depend on the mass of the test object.

Transcript

00:01 A spaceship of mass m travels from earth to the moon along a line that passes through the center of the earth and the center of the moon.

00:08 Here i've drawn a diagram showing the spaceship traveling from earth to the moon in a straight line, and the distance between the center of the earth and the moon is 3 .84 times 10 to the 8 meters.

00:22 Here i've also written on the side the mass of the earth and the mass of the moon.

00:27 The problem wants us to find at what distance from the center of the earth is the force.

00:31 Due to the earth twice as strong as the force due to the moon.

00:36 So let's just go ahead and write that in a math equation.

00:40 So force of the earth on the spaceship is equal to twice the magnitude of the force from the moon on the spaceship.

00:55 So recall that f1 .2 is equal to big g, m1, m2 divided by r squared.

01:06 So we can go ahead and plug that formula in over here.

01:16 So g.

01:19 M.

01:19 E.

01:21 Ms.

01:22 Which we're told that ms is just m for mass.