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Chapter 12: Problem 62

Newton was holding an apple of mass \(100 . \mathrm{g}\) and thinking about the gravitational forces exerted on the apple by himself and by the Sun. Calculate the magnitude of the gravitational force acting on the apple due to (a) Newton, (b) the Sun, and (c) the Earth, assuming that the distance from the apple to Newton's center of mass is \(50.0 \mathrm{~cm}\) and Newton's mass is \(80.0 \mathrm{~kg}\).

Short Answer

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#tag_title# Question Calculate the magnitudes of the gravitational forces acting on a 100g apple due to Newton, the Sun, and the Earth.

Step by step solution

01

Calculate the gravitational force due to Newton

F_N = G * (m1 * m_N) / d_N^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 80.0kg) / (0.5m)^2. ## Part (b): Gravitational force due to the Sun ##
02

Calculate the gravitational force due to the Sun

F_s = G * (m1 * m_s) / d_s^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 1.989 x 10^30 kg) / (1.496 x 10^11 m)^2. ## Part (c): Gravitational force due to the Earth ##
03

Calculate the gravitational force due to the Earth

F_E = G * (m1 * m_E) / d_E^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 5.972 x 10^24 kg) / (6.371 x 10^6 m)^2. After calculating the gravitational force for each case using the given values and the formula, we get the magnitudes of the gravitational forces acting on the apple due to Newton, the Sun, and the Earth.

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Most popular questions from this chapter

A 1000.-kg communications satellite is released from a space shuttle to initially orbit the Earth at a radius of \(7.00 \cdot 10^{6} \mathrm{~m}\). After being deployed, the satellite's rockets are fired to put it into a higher altitude orbit of radius \(5.00 \cdot 10^{7} \mathrm{~m} .\) What is the minimum mechanical energy supplied by the rockets to effect this change in orbit?

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Determine the minimum amount of energy that a projectile of mass \(100.0 \mathrm{~kg}\) must gain to reach a circular orbit \(10.00 \mathrm{~km}\) above the Earth's surface if launched from (a) the North Pole or from (b) the Equator (keep answers to four significant figures). Do not be concerned about the direction of the launch or of the final orbit. Is there an advantage or disadvantage to launching from the Equator? If so, how significant is the difference? Do not neglect the rotation of the Earth when calculating the initial energies.

A satellite is in a circular orbit around a planet. The ratio of the satellite's kinetic energy to its gravitational potential energy, \(K / U_{\mathrm{g}}\), is a constant whose value is independent of the masses of the satellite and planet, and of the radius and velocity of the orbit. Find the value of this constant. (Potential energy is taken to be zero at infinite separation.)

A satellite of mass \(m\) is in an elliptical orbit (that satisfies Kepler's laws) about a body of mass \(M,\) with \(m\) negligible compared to \(M\) a) Find the total energy of the satellite as a function of its speed, \(v\), and distance, \(r\), from the body it is orbiting. b) At the maximum and minimum distance between the satellite and the body, and only there, the angular momentum is simply related to the speed and distance. Use this relationship and the result of part (a) to obtain a relationship between the extreme distances and the satellite's energy and angular momentum. c) Solve the result of part (b) for the maximum and minimum radii of the orbit in terms of the energy and angular momentum per unit mass of the satellite. d) Transform the results of part (c) into expressions for the semimajor axis, \(a\), and eccentricity of the orbit, \(e\), in terms of the energy and angular momentum per unit mass of the satellite.
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