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Chapter 8: Problem 4

The force of gravity on an object is proportional to the object's mass, yet all objects fall with the same gravitational acceleration. Why?

Short Answer

Expert verified
All objects fall with the same gravitational acceleration because the acceleration experienced by a falling object, per Newton's Second Law of Motion and the Universal Law of Gravitation, is equal to the acceleration due to gravity. The mass of the object drops out of the calculation, leading to a constant acceleration for any object falling in a vacuum near the surface of the earth.

Step by step solution

01

Understand Newton's Second Law of Motion

Newton's Second Law of Motion states that the force on an object is equal to its mass multiplied by its acceleration. This can be represented as \(F = ma\). In the context of gravity, the force is the weight of the object, which is the product of the object's mass and the acceleration due to gravity (\(g\)). This can be represented as \(F = mg\).
02

Understand the Universal Law of Gravitation

The Universal Law of Gravitation states that the force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This can be represented as \(F = G \frac{m1m2}{r^2}\), where \(G\) is the gravitational constant, \(m1\) and \(m2\) are the masses of the two objects and \(r\) is the distance between them. On the surface of the earth, where the distance to the center of the earth \(r\) is essentially constant, this simplifies to \(F = mg\) where \(m\) is the mass of the object and \(g = G \frac{M_{earth}}{r^2}\) is the acceleration due to gravity.
03

Analyze the Relationship between Force, Mass, and Acceleration

Combing Newton's second law (\(F = ma\)) and the formula for gravitational force (\(F = m*g\)), we can equate the two force equations and solve for acceleration (\(a\)): \[ma = mg\] This simplifies to: \[a = g\] Therefore, the acceleration experienced by a falling object is equal to the acceleration due to gravity, irrespective of the mass of the object. All objects fall with the same acceleration (neglecting air resistance) as the mass \(m\) cancels out in the equation.

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Most popular questions from this chapter

A projectile is launched vertically upward from a planet of mass \(M\) and radius \(R ;\) its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance \(r\) from the planet's center.

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