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Chapter 5: Problem 3

As Pluto moves from the perihelion to the aphelion. the work done by gravitational pull of Sun on Pluto is (1) is zero (2) is positive (3) is negative (4) depends on sense of rotation

Short Answer

Expert verified
The work done is negative (option 3).

Step by step solution

01

Understanding Perihelion and Aphelion

Perihelion is the point in the orbit of a celestial body, like Pluto, where it is closest to the Sun. Aphelion is the point where it is farthest from the Sun. As Pluto moves from perihelion to aphelion, it moves against the Sun's gravitational pull.
02

Analyzing Gravitational Work

The work done by a force when it causes displacement can be calculated by the formula \( W = F \cdot d \cdot \cos(\theta) \). In the context of celestial bodies, gravitational force is conservative, meaning the work done over a complete orbit is zero. However, when just considering the segment from perihelion to aphelion, the displacement is against the force's direction.
03

Determining the Sign of the Work Done

When a body moves against the force of gravity, the work done by the gravitational force is negative because \( \theta = 180^\circ \) and \( \cos(\theta) = -1 \). In this specific movement from perihelion to aphelion, the gravitational force tries to pull Pluto closer (inwards), but it moves outward.

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Key Concepts

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

Gravitational Force
Gravitational force is a fundamental interaction that draws two masses toward each other. For any two objects, the gravitational force acting on them is proportional to the product of their masses and inversely proportional to the square of the distance between them. This classical idea is captured by Newton's Law of Universal Gravitation, given by the formula: \[ F = G \frac{m_1 m_2}{r^2} \]where:
  • \( F \) is the gravitational force between the objects,
  • \( G \) is the gravitational constant,
  • \( m_1 \) and \( m_2 \) are the masses of the two objects,
  • \( r \) is the distance between the centers of their masses.
Gravitational force acts as a central force directed along the line joining the centers of the masses.
This simple model explains a multitude of phenomena in our universe, from the fall of an apple on Earth to the orbit of planets around the sun. The gravitational force working between celestial bodies like planets and their stars is what keeps planets in orbit.
Perihelion and Aphelion
When discussing orbits, perihelion and aphelion are key terms to understand. These terms describe specific points in an elliptical orbit where a body is nearest or farthest from the Sun, respectively.
  • Perihelion: This is the point in a planet's orbit when it is closest to the Sun. At the perihelion, the gravitational pull is strongest and the planet moves fastest in its orbit due to the stronger gravitational attraction.
  • Aphelion: This is the opposite, where the planet is farthest from the Sun. The gravitational pull here is weaker compared to perihelion, causing the planet to move at its slowest speed in its orbit.
Understanding these points is crucial when analyzing how gravitational force affects the motion of planets. For Pluto, its movement from perihelion to aphelion involves moving outward against the Sun's gravitational pull, reflecting specific gravitational dynamics.
Conservative Forces in Physics
A conservative force is a type of force where the work done in moving a particle between two points is independent of the path taken. The most familiar example of such a force is gravity.
  • For a conservative force, the total work done around any closed path is zero. This is a key property and it's precisely why no net work is done over a complete orbit.
  • When considering parts of the orbit specifically, such as Pluto's movement from perihelion to aphelion, the work done by gravity is negative.
  • This happens because Pluto moves against the gravitational pull, meaning the work done is in opposition to the force direction.
In practical terms, understanding that gravitational force is conservative allows us to conserve energy calculations in celestial mechanics. This makes it possible to predict planetary motions over long timescales without concerning path details.

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Most popular questions from this chapter

Two satellites of the same mass are launched in the same orbit around the earth so as to rotate opposite to each other. If they collide inelastically and stick together as wreckage. the total energy of the system just after collision is (1) \(-\frac{2 G M m}{r}\) (2) \(-\frac{G M m}{r}\) (3) \(\frac{G M m}{2 r}\) (4) \(\frac{G M m}{4 r}\)

Which of the following are correct? (1) An astronaut going from the earth to the Moon will experience weightlessness once. (2) When a thin uniform spherical shell gradually shrinks maintaining its shape, the gravitational potential at its centre decreases. (3) In the case of a spherical shell, the plot of \(V\) versus \(r\) is continuous. (4) In the case of a spherical shell, the plot of gravitational field intensity \(I\) versus \(r\) is continuous.

If three particles, each of mass \(M\), are placed at the three corners of an equilateral triangle of side \(a\), the forces exerted by this system on another particle of mass \(M\) placed (i) at the midpoint of a side and (ii) at the centre of the triangle are, respectively, (1) \(0, \frac{4 G M t}{3 a^{2}}\) (2) \(\frac{+G \| F}{3 a^{2}}, 0\) (3) \(3 \frac{3 G M^{2}}{a^{2}}, \frac{G M t^{2}}{a^{2}}\) (4) 0,0

Choose the correct statements from the following: (I) The magnitude of the gravitational force between two bodies of mass \(1 \mathrm{~kg}\) each and separated by a distance of \(1 \mathrm{~m}\) is \(9.8 \mathrm{~N}\) (2) The higher the value of the escape velocity for a planet, the higher is the abundance of the lighter gases in its atmosphere. (3) The gravitational force of attraction between two bodies of ordinary mass is not noticeable because the value of the gravitational constant is extremely small. (4) Force of friction arises due to gravitational attraction.

Two particles of mass \(m\) and \(4 m\) are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If \(G\) is the universal gravitational constant. Then at separation \(r\), (Assume zero reference potential energy at infinite separation) (1) the total energy of the two object is zero (2) their relative velocity of approach is \(\left(\frac{10 G m}{r}\right)^{1 / 2}\) in magnitude (3) the total kinetic energy of the object is \(\frac{4 G m^{2}}{r}\) (4) Net angular momentum of both the particles is zero about any point
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