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.
part 2 of 3 In 3d, we may extend this idea to cylindrical coordinates (r, o, z), which you may think of as a plane polar coordinate system with the z-dimension tacked on. (r, 6,2) y This coordinate system is useful when deal- ing with problems that have radial symmetry about some central axis. Question: You are given a hollow cylinder of radius R whose central axis is the z-axis and whose base rests on the xy-plane. What are the Cartesian coordinates of an arbitrary point on the surface of the cylinder in terms of cylindrical quantities? 1. (x² + y², tan¯ Y ¹(²-).. Т 2. (R cos o, R sin o, z) ○ 3. (x, y, z) ○ 4. (R, o, z) 5. (x cos o, y sin o, z) O 6. (R sin o, R cos , z)
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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.
Problem 1 Consider two celestial bodies of masses m₁ and m2. The bodies only interact gravitationally and can be considered pointlike for the purpose of this problem. (a) List the conserved quantities of the system. (b) For each conserved quantity, state the Noether symmetry responsible for its conservation.
This is for a solid sphere of radius R rolling down a hemisphere of radius 5R. please help me understand how to produce these 3 equations. Thank you! the holonomic constraint equations are g₁ (r, 0, 0) = r — 6R = 0 and 92 (r, 0, 0) = Ro-5R0=0 L (r, 0, 0) = { mr² + 1/{mr²0² + / mR²² - mgrcos (0) produce three Euler-Lagrange equations. SL d (4) - 500 dt SL (S) d dt d ($) dt - - 8g2 +1₂50/200 dgi Σi dø δὲ №₁0% + A2 1 δL = λ + λ = Σ SL 80 80 sgt 80 8gi SL 61 = A₁ / + √₂/² = Σ₁ %fh λ dr dr
A particular point P on a rigid object has co-ordinates (x, y, z) = (2.0,2.0,0). The object is now rotated about the origin according to the Euler angles (01 , 02, 03) = (-45°, 60°, 70°) and also displaced -2.0 units along the y-axis What are the new co-ordinates of the point P?
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