Solve it asap possible a) Given two points with coordinates A = (XA, YA) and B = (XB,YB) in R² in a certainCartesian system, we define the Euclidean distance between them as (Ar)² = (Ax)² +(Ay)², where Ax = xB - Xд and Aу = уB - YA. Show that the Euclidean distance isinvariant to rotations, that is, (Ar)² = (Ar')² where the relationship between bothcoordinate system isWith a constant angle.x'x cos y sin=y' x sin 0 + y cos 0

Step 1:To show that the Euclidean distance is invariant under revolution, we really want to exhibit…

Answered: a) Given two points with coordinates A…

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter8: Central-force Motion

Section: Chapter Questions

Problem 8.18P

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Compute the gravitational attraction on a unit mass at the origin due to the mass (of constant density) occupying the volume inside the sphere r = 2a and above the plane z = a. Hint: The magnitude of the gravitational force on the unit mass due to the element of mass dM at (r, θ, φ) is (G/r2)dM. You want the z component of this since the other components of the total force are zero by symmetry. Use spherical coordinates.
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Consider a circular orbit in the Schwarzschild spacetime. We take the orbit to lie on the plane θ = π/2. From the radial geodesic equation, find an expression for (dϕ/dt)2, and verify that it reproduces Kepler’s third law of planetary motion.
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The Kentucky Derby is a 2 km race. Assume the track is circular and horses run counterclockwise around the track starting due east of center. Let theta be the angle (in radians) the horse has swept out counterclockwise since starting the race. 1)How many kilometers has the horse traveled if the angle swept out by the horse is pi radians? 90 degrees? Determine which is longer and explain your answers. 2)Suppose the horse travels at a constant speed of 55 km per hour. How long does it take the horse to run once around the track? Justify and include units.
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1. Consider a vertical plane in a constant gravitational field. Let the origin of a coordinate system be located at some point in this plane. A particle of mass m moves in the vertical plane under the influence of gravity and under the influence of an addition force f = -Ar"- directed toward the origin (r is the distance from the origin; A and a [#0 or 1] are constants). Choose appropriate generalized coordinates, and find the Lagrangian equations of motion (don't solve it). Is the angular momentum about the origin conserved? Explain.

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