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Chapter 5: Problem 2

The earth is attracted to an object with a force equal and opposite to the force of the earth on the object. If this is true, why is it that when you drop an object, the earth does not have an acceleration equal and opposite to that of the object?

Short Answer

Expert verified
The Earth's acceleration is negligible due to its massive size compared to the object's.

Step by step solution

01

Define Newton's Third Law

Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. This means that the force the Earth exerts on an object is equal in magnitude and opposite in direction to the force the object exerts on the Earth.
02

Identify Forces and Interaction Pairs

When an object is dropped, it is pulled towards the Earth due to gravitational force. Simultaneously, the object exerts an equal and opposite force on the Earth. These forces form an interaction pair as per Newton's Third Law.
03

Understand Mass Difference

The Earth has an enormous mass compared to the object. The mass of the Earth is approximately \(6 \times 10^{24}\) kg, while the mass of a typical object like an apple is about 0.1 kg. This large difference in mass is crucial to understanding their resulting accelerations.
04

Apply Newton's Second Law

Newton's Second Law states that force equals mass times acceleration, \( F = ma \). When we apply this to the forces acting between the Earth and the object, we know the force exerted is the same, but the accelerations will be different due to the difference in masses.
05

Calculate or Theorize Object and Earth's Acceleration

If \( F = ma \), then the acceleration \( a \) of the object is \( F/m_{object} \). The acceleration of Earth would be \( F/m_{earth} \). Given the large mass of the Earth, its calculated acceleration is practically zero, and thus, unnoticeable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Third Law
Newton's Third Law of Motion is a fundamental principle of physics that helps us understand the interactions between two bodies. It essentially states that for every action, there is an equal and opposite reaction. In the context of dropping an object, when the Earth exerts a gravitational force on the object, the object simultaneously exerts a gravitational force of equal magnitude but in the opposite direction on the Earth. This means that forces always come in pairs known as "action-reaction pairs." These pairs are equal and opposite, so if the Earth pulls an apple downward, the apple pulls the Earth upward with an equal force.
Newton's Second Law
To grasp why the Earth's acceleration isn't noticeable when you drop something, we turn to Newton's Second Law of Motion. This law expresses the relationship between force, mass, and acceleration as: \( F = ma \)where
  • \( F \) is the force applied,
  • \( m \) is the object's mass,
  • \( a \) is the acceleration produced.
When a force is applied to an object, the resulting acceleration depends on the object's mass. For a small object like an apple, the gravitational force from the Earth results in a noticeable acceleration towards the Earth. Conversely, the same force exerted on the massive Earth results in a negligible acceleration, in accordance to the formula \( F = ma \). The massive Earth requires much more force to achieve even a small acceleration.
Gravitational Force
Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another. For any two masses, like the Earth and an apple, there exists a gravitational attraction. Formally, the gravitational force acting between two masses can be described by the equation:\[ F = \frac{G m_1 m_2}{r^2} \]where
  • \( F \) is the gravitational force between the masses,
  • \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{N}\, \text{(m/kg)}^2) \),
  • \( m_1 \) and \( m_2 \) are the masses, and
  • \( r \) is the distance between the centers of the two masses.
Even though this force is mutual, the result of dropping an object is far more significant on the object than on the Earth due to their vast difference in mass.
Mass and Acceleration
The concepts of mass and acceleration are intricately linked through Newton's Second Law of Motion. In simple terms, mass refers to the amount of matter in an object and is a measure of its inertia, or resistance to changes in motion. Acceleration, on the other hand, is the rate at which an object changes its velocity. As we observe the interaction between the Earth and an object being dropped, we see this relationship in action:
  • A small mass, like an apple, undergoes significant acceleration due to Earth's gravitational pull because of its relatively small inertia. This is why we see the apple quickly accelerate towards the ground.
  • The Earth, with its enormous mass, accelerates imperceptibly because the same amount of force causes a tiny change in motion due to its huge inertia.
This difference in mass is key to understanding why we don't see the Earth's movement even as it attracts objects towards it.

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Most popular questions from this chapter

Mountain climbers with masses \(m\) and \(M\) are roped together while crossing a horizontal glacier when a vertical crevasse opens up under the climber with mass \(M\). The climber with mass \(m\) drops down on the snow and tries to stop by digging into the snow with the pick of an ice ax. Alas, this story does not have a happy ending, because this doesn't provide enough friction to stop. Both \(m\) and \(M\) continue accelerating, with \(M\) dropping down into the crevasse and \(m\) being dragged across the snow, slowed only by the kinetic friction with coefficient \(\mu_{k}\) acting between the ax and the snow. There is no significant friction between the rope and the lip of the crevasse. (a) Find the acceleration \(a\). (b) Check the units of your result. (c) Check the dependence of your equation on the variables. That means that for each variable, you should determine what its effect on \(a\) should be physically, and then what your answer from part a says its effect would be mathematically.

Ginny has a plan. She is going to ride her sled while her dog Foo pulls her, and she holds on to his leash. However, Ginny hasn't taken physics, so there may be a problem: she may slide right off the sled when Foo starts pulling. (a) Analyze all the forces in which Ginny participates, making a table as in section \(5.3\). (b) Analyze all the forces in which the sled participates. (c) The sled has mass \(m\), and Ginny has mass \(M\). The coefficient of static friction between the sled and the snow is \(\mu_{1}\), and \(\mu_{2}\) is the corresponding quantity for static friction between the sled and her snow pants. Ginny must have a certain minimum mass so that she will not slip off the sled. Find this in terms of the other three variables. (d) Interpreting your equation from part c, under what conditions will there be no physically realistic solution for \(M ?\) Discuss what this means physically.

A little old lady and a pro football player collide head-on. Compare their forces on each other, and compare their accelerations. Explain.

Pick up a heavy object such as a backpack or a chair, and stand on a bathroom scale. Shake the object up and down. What do you observe? Interpret your observations in terms of Newton's third law.

When you stand still, there are two forces acting on you, the force of gravity (your weight) and the normal force of the floor pushing up on your feet. Are these forces equal and opposite? Does Newton's third law relate them to each other? Explain.
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