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Chapter 7: Problem 4

The force from the Sun on the Earth acts directly towards the Sun, yet the Earth does not fall into the Sun. Explain.

Short Answer

Expert verified
The gravitational force from the Sun provides the centripetal force necessary for Earth's orbit, countering Earth's inertia and preventing it from falling into the Sun.

Step by step solution

01

Understand the Gravitational Force

The Sun exerts a gravitational force on the Earth that pulls it towards the Sun. This force acts along the line connecting the centers of the Sun and the Earth.
02

Consider Earth's Motion

The Earth is not stationary; it orbits around the Sun. This motion is tangential to the direction of the gravitational force exerted by the Sun.
03

Centripetal Force and Circular Motion

The gravitational force provides the necessary centripetal force that keeps the Earth in its nearly circular orbit. Despite the Sun’s gravitational pull, Earth's inertia causes it to move forward in its orbit.
04

Newton's First Law and Orbital Motion

According to Newton's First Law of Motion, an object in motion will stay in motion unless acted upon by an external force. The gravitational pull from the Sun continuously deflects Earth's path, resulting in an elliptical orbit rather than a straight line towards the Sun.

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Key Concepts

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

Orbital Motion
The motion of Earth around the Sun is a perfect example of orbital motion. Imagine Earth's path as a loop or an oval around the Sun. Even though the gravitational force from the Sun is pulling Earth towards it, the planet's speed and direction keep it moving in this steady path.
Orbital motion involves a balance between two things:
  • The gravitational pull from the Sun
  • The forward motion of the Earth

This balance ensures that instead of falling directly into the Sun, Earth follows an orbit. If Earth's speed increased, it would move to a higher orbit. If it slowed down, it would move to a closer one.
Centripetal Force
Centripetal force is the force that keeps an object moving in a curved path and is directed towards the center of the path. For Earth, this force comes from the gravitational pull of the Sun.
Because of this force:
  • The Earth is kept in its orbit
  • It doesn't just fly off into space

In mathematical terms, the centripetal force \( F_c \) needed to keep an object of mass \( m \) in a circular orbit of radius \( r \) moving at a velocity \( v \) is given by \[ F_c = \frac{mv^2}{r} \]
This equation helps explain why faster-moving objects (those with a high velocity \( v \)) need more centripetal force to stay in a circular path.
Newton's First Law
Newton's First Law of Motion states: 'An object in motion will remain in motion unless acted upon by an external force.'
For the Earth and other planets, this law applies perfectly.
  • The Earth's inertia wants to keep it moving in a straight line.
  • The Sun's gravity pulls Earth towards the Sun.

The combination of these two effects causes Earth to move in an elliptical orbit instead of falling into the Sun. Without the Sun's gravity (the external force), Earth would just move off in a straight line into space.
This law helps us understand why planets orbit stars and not just drift away or fall directly into them.

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Most popular questions from this chapter

A soccer player is making a free kick and the opposing team is making a wall to protect their goal. Is it always theoretically possible for the kicker to hit the goal in an ideal situation with only a vertical acceleration due to gravity?

In this project we will develop a model to determine the motion of a weather balloon released from the ground. We start from a simplified model and gradually add features to make to model more realistic. After the balloon is released, it is driven by buoyancy. Initially, we will assume that the buoyancy force is a constant, \(B\). (a) Draw a free-body diagram of the balloon. Identify the forces and introduce symbols. Indicate the relative magnitudes of the forces by the length of the vectors. First, let us neglect air resistance. (b) What is the acceleration of the balloon? (c) Find the position and velocity of the balloon as a function of time. Let us now introduce air resistance, using a quadratic law: \(\mathbf{F}_{D}=-D v \mathbf{v}\). (d) Show that the acceleration of the balloon in the upward \((z)\) direction is \(a_{z}=\) \((B / m)-g-(D / m)\left|v_{z}\right| v_{z}\), where \(v_{z}\) is the velocity in the \(z\)-direction. (e) Sketch the acceleration and velocity as a function of time for the model including air resistance. (f) Find the asymptotic (terminal) velocity of the balloon. The balloon is released on a windy day, with a wind blowing with a velocity \(\mathbf{w}=w \mathbf{i}\) along the horizontal \(x\)-axis. (g) How does the wind modify the air resistance force \(\mathbf{F}_{D}\) on the balloon? (h) Draw a free-body diagram for the balloon in this case. Indicate the magnitude of the forces by the relative lengths of the vectors. (i) Find an expression for the acceleration a of the balloon. What are the initial conditions for the motion of the balloon? (j) Why do we call the motion in the \(z\) and the \(x\) direction "coupled" in this case? Can you determine the motion of the balloon analytically? (k) We can determine the motion of the balloon using numerical methods. Write a program to find the velocity \(\mathbf{v}(t)\) and position \(\mathbf{r}(t)\) as functions of time. (It is sufficient to only include the main integration step in your answer - that is the part that determines \(\mathbf{v}(t+\Delta t)\) and \(\mathbf{r}(t+\Delta t)\) given \(\mathbf{v}(t)\) and \(\mathbf{r}(t)\). Figure \(7.16\) shows the result of a simulation. (l) Describe the motion of the balloon. Illustrate by relevant sketches. \((\mathbf{m})\) Find the asymptotic (terminal) velocity of the balloon. In a real situation, the wind velocity is smaller near the ground and increases gradually to the full velocity \(w_{0}\) as the balloon moves upward. Typically, the velocity of the wind can be described by \(\mathbf{w}=w_{0}(1-\exp (-z / d)) \mathbf{i}\), where \(d=10 \mathrm{~m}\) is a length determining the transition. (n) Rewrite your program to include this effect. (o) What is the terminal velocity of the balloon now?

You tie a long, strong rope between your car and a tree in order to exert a large force on the car. How can you pull the rope to ensure that you pull at the car with a much larger force than you can exert on the rope?
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