The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M. For all parts of this problem, where appropriate, use G for the universal gravitational constant. Find L, the magnitude of the angular momentum of the satellite with respect to the center of the planet. Express your answer in terms of m, M, G, and R.   The quantities v, K, U, and Lall represent physical quantities characterizing the orbit that depend on radius R. Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answer as a comma-separated list of exponents corresponding to v, K, U, and L, in that order. For example, -1,-1/2,-0.5,-3/2 would mean v∝R−1∝−1, K∝R−1/2∝−1/2, and so forth.

Q: A tunnel is drilled from the surface of a planet straight down to its antipodal point on the other…

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Q: The asteroid 243 Ida has a mass of about 4.0×10¹⁶ kg and an average radius of about 16 km (it's not…

A: Mass , m = 4.0 x 1016 kgRadius , R = 16 km = 16 x 103 mTo find = a) launch speed of rock .…

Q: A satellite is orbiting around a planet in a circular orbit. The radius of the orbit, measured from…

A: R=1.8×107 mM=4.8×1024 kg From the conservation of energy, 12mv2=GMm2R, here G is gravitational…

Q: Assume that the box and disk each have mass m, the top of the incline is at height h, and the angle…

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Q: At apoapsis (furthest point from the star), the

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Q: The code Orbit (Appendix J) can be used to generate orbital positions, given the mass of the central…

A: Required : Plots of the orbits.

Q: a) A 95 kg astronaut stands on the surface of a spherical asteroid with a mass of 4.5 x 1015 kg and…

A: Please see the answer below.

Q: A conical pendulum consists of a mass, m, at the end of a length of an ideal rope of length l that…

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Q: You are a scientist exploring a mysterious planet. You have performed measurements and know the…

A: Given information:Radius of the planet (R) = dFree fall acceleration of the planet = a“G” is…

Q: A ring of radius 6.5 m lies in the x-y plane, centered on the origin. The portions of the ring in…

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Q: The attractive gravitational force between a spherical body of mass M and another spherical body of…

A: ω=2πf  Acceleration of satellite = ω2r r = radius of revolution = RE+h  Force experienced by…

Q: Assume that each planet moves in an orbit given by the vectorvalued function r. Let r = || r||, let…

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Q: Very far from earth (at R=∞), a spacecraft has run out of fuel and its kinetic energy is zero. If…

A: Solution:Explanation:Mechanical energy includes both kinetic energy (energy due to motion) and…

Q: To find some of the parameters characterizing an object moving in a circular orbit. The motivation…

A: Gravity is one of the fundamental forces of nature. It is the weakest of them all and is always…

Q: Consider a situation where a satellite is orbiting Mars at a distance equal to 3 times the radius of…

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Q: The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's orbit.…

A: The acceleration due to gravity at Neptune due to Pluto can be obtained as…

Q: Suppose the shuttle is fired into orbit and 14 minutes later it reaches its farthest distance north…

A: discuss the problem by the simple equation of wave and draw the graph find answer.

Q: In the x-y plane there is an infinitely thin layer of uniform density Y2(unit: kg/m²), and on the…

A: We can use gauss law to find the total gravitational field. It is given by, ∮E→.dA→=-4πGMenclosed…

Q: For an elliptic Earth orbit find the ratio of the time required to travel from a point on the lower…

A: Given: The value of ν for lower latus parameter is -π3. The value of ν for upper latus parameter is…

Q: A planet moves in an elliptical orbit around a star. Vp is the component of the planet's linear…

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Q: M x= -G +G M x2 (x — L)²' -

A: Given the equation d2xdt2=−GMx2+GM(x−L)2⇒d2xdt2=−GMx2+GM(L−x)2    [As (x-L)2=(L-x)2]This is a…

Q: Consider the earth following its nearly circular orbit (dashed curve) about the sun.(Figure 2) The…

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Q: Describe the motion of a particle on the rim of a rolling wheel of radius a, given by the position:…

A: Motion of particle on the rim of a wheel is given asx(t) = a(θ-sinθ)y(t) = a(1-cosθ)here θ = ω×tThe…

Q: onsider a planet consisting of two thin concentric shells, as in the picture, both made from the…

A: The shell theorem states that a spherically body affects external objects gravitationally as if all…

Q: You are a scientist exploring a mysterious planet. You have performed measurements and know the…

A: let the mass of the planet be m and that of the star is M,the weight of an object at the surface of…

Q: In the time of Johannes Kepler, it was believed that the orbit of Earth was circular, whereas the…

A: Kepler knew the geometry of ellipses very well, and recognized that this information made it quite…

Q: A number of gas giant planets orbiting other stars at distances less than 1 A.U. have been…

A: Since you have asked multiple questions, I am answering the first 3 subparts. If you need answers of…

Q: A number of gas giant planets orbiting other stars at distances less than 1 A.U. have been…

A: Hey there, since you have asked multiple questions we have solved the first three for you. Please…

Q: Stars and black holes in a binary system orbit each other in circular orbits of radius r1 and r2…

A: mass of an orbital m= 1.98x1030 kg v1 =5.36 times faster than our Sun T=30 hours r1=?,r2=?m2=?

Q: Five small spheres each with a mass of 48 kg are attached to the vertices of a pentagon. A set of…

A: The formula for the gravitational field is given by the following expression, F=-GMr2 where M is the…

Q: An object of mass m is dropped from height h above a planet of mass M and radius R. Find an…

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Q: Consider a planet in orbit around a star as depicted below.  It begins at position A, goes to B…

A: (a)Using Kepler’s laws, rank the positions in terms of planetary speed, slowest to fastest and give…

Q: A roller coaster of mass m (when filled with passengers) goes over the crest öf å čircular section…

A: a) As we know that the force equation for the motion is :Fg-N=mv2rnow, its known that F=mg,…

Q: Starting from rest, a child throws a ball of mass m with an initial speed v, at an angle B with the…

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Q: A geosynchronous orbit for a satellite is a circular orbit about the equator of the Earth in which…

A: The net  force acting on the satellite Fg=FcFg= gravitational forceFc=centripetal…

Q: A near-Earth orbit (NEO) is a circular orbit at a height above the surface of the Earth that is very…

A: These laws explain how a satellite stays in orbit.Law (1): A satellite would tend to go off in a…

Q: A particle of mass m is placed at a distance r away from the center of a thin circular hoop of mass…

A: Given, mass as, m1=Mm2 =m radius=R, distance=r.

Q: In 2014, the Rosetta space probe reached the comet Churyumov– Gerasimenko. Although the comet’s core…

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Q: A rocket in free flight is approaching a planet of mass M as shown above.  Free flight means that…

A: It is given that, radius of planet is r1=r=54×105 m, mass of the planet, M=1025 kg, distance of…

Q: Find an expression for the kinetic energy of a satellite of mass m in an orbit of radius r about a…

A: Consider a satellite having mass m revolving around the planet with mass M and the radius of the…

Q: The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's orbit.…

A: Given:-  The existence of the dwarf planet Pluto was proposed based on irregularities in Neptune's…

Q: Problem 1: A ball of mass m attached to a string is being swung around in a horizontal circle of…

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Central Force

Central force is the force that acts along the center of mass of one body to the center of mass of another body. Its direction from or away from a point on a body to another is called the center of the force. Such forces include gravitational force between two massive objects or electrostatic forces between two charged particles. So from atoms to planets, the central forces play an important role in shaping our daily lives. One of the remarkable features of central forces is that they are conservative, i.e. the total energy of a central force system can always be expressed as a gradient of a potential energy function.