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Chapter 12: Problem 17

Is the orbital speed of the Earth when it is closest to the Sun greater than, less than, or equal to the orbital speed when it is farthest from the Sun? Explain.

Short Answer

Expert verified
Answer: The orbital speed of Earth when it is closest to the Sun (perihelion) is greater than when it is farthest from the Sun (aphelion).

Step by step solution

01

Understand the concept of orbital speed

Orbital speed is the speed at which an object (in this case, the Earth) moves in its orbit around another object (the Sun). The primary factor affecting the orbital speed of an object is the gravitational force acting between the objects, which depends on their masses and the distance between their centers.
02

Apply Kepler's second law

Kepler's second law, also known as the Law of Equal Areas, states that a line joining a planet and its star sweeps out equal areas in equal periods of time. This law implies that when the distance between the planet and the star is reduced, the planet moves faster to cover the same area in the same time.
03

Determine the orbital speed at perihelion and aphelion

We need to find the Earth's orbital speed when it is closest to the Sun (perihelion) and when it is farthest from the Sun (aphelion). According to Kepler's second law, Earth moves faster at perihelion than at aphelion because of the different distances between Earth and Sun.
04

Compare the orbital speeds at perihelion and aphelion

Since Earth moves faster when it is closer to the Sun (perihelion), we can conclude that the orbital speed of Earth when it is closest to the Sun is greater than when it is farthest from the Sun.

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Most popular questions from this chapter

Newton's Law of Gravity specifies the magnitude of the interaction force between two point masses, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) as \(F(r)=G m_{1} m_{2} / r^{2} .\) The gravitational constant \(G\) can be determined by directly measuring the interaction force (gravitational attraction) between two sets of spheres by using the apparatus constructed in the late 18th century by the English scientist Henry Cavendish. This apparatus was a torsion balance consisting of a 6.00 -ft wooden rod suspended from a torsion wire, with a lead sphere having a diameter of 2.00 in and a weight of 1.61 lb attached to each end. Two 12.0 -in, 348 -lb lead balls were located near the smaller balls, about 9.00 in away, and held in place with a separate suspension system. Today's accepted value for \(G\) is \(6.674 \cdot 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) Determine the force of attraction between the larger and smaller balls that had to be measured by this balance. Compare this force to the weight of the small balls.

Two 30.0 -kg masses are held at opposite corners of a square of sides \(20.0 \mathrm{~cm} .\) If one of the masses is released and allowed to fall toward the other mass, what is the acceleration of the first mass just as it is released? Assume that the only force acting on the mass is the gravitational force of the other mass. a) \(1.5 \cdot 10^{-8} \mathrm{~m} / \mathrm{s}^{2}\) b) \(2.5 \cdot 10^{-8} \mathrm{~m} / \mathrm{s}^{2}\) c) \(7.5 \cdot 10^{-8} \mathrm{~m} / \mathrm{s}^{2}\) d) \(3.7 \cdot 10^{-8} \mathrm{~m} / \mathrm{s}^{2}\)

Three asteroids, located at points \(P_{1}, P_{2},\) and \(P_{3}\), which are not in a line, and having known masses \(m_{1}, m_{2}\), and \(m_{3}\), interact with one another through their mutual gravitational forces only; they are isolated in space and do not interact with any other bodies. Let \(\sigma\) denote the axis going through the center of mass of the three asteroids, perpendicular to the triangle \(P_{1} P_{2} P_{3} .\) What conditions should the angular velocity \(\omega\) of the system (around the axis \(\sigma\) ) and the distances $$ P_{1} P_{2}=a_{12}, \quad P_{2} P_{3}=a_{23}, \quad P_{1} P_{3}=a_{13} $$ fulfill to allow the shape and size of the triangle \(P_{1} P_{2} P_{3}\) to remain unchanged during the motion of the system? That is, under what conditions does the system rotate around the axis \(\sigma\) as a rigid body?

Two planets have the same mass, \(M .\) Each planet has a constant density, but the density of planet 2 is twice as high as that of planet \(1 .\) Identical objects of mass \(m\) are placed on the surfaces of the planets. What is the relationship of the gravitational potential energy, \(U_{1},\) on planet 1 to \(U_{2}\) on planet \(2 ?\) a) \(U_{1}=U_{2}\) b) \(U_{1}=\frac{1}{2} U_{2}\) c) \(U_{1}=2 U_{2}\) d) \(U_{1}=8 U_{2}\) e) \(U_{1}=0.794 U_{2}\)

A planet with a mass of \(7.00 \cdot 10^{21} \mathrm{~kg}\) is in a circular orbit around a star with a mass of \(2.00 \cdot 10^{30} \mathrm{~kg} .\) The planet has an orbital radius of \(3.00 \cdot 10^{10} \mathrm{~m}\). a) What is the linear orbital velocity of the planet? b) What is the period of the planet's orbit? c) What is the total mechanical energy of the planet?
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