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- The quantities v, K, U, and L all 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 ∞ R¹, K & R-¹/2, and so forth.The triangle drawn in question 2 (with v and Δv) is similar to the triangle drawn in question 3 (with r and the straight line distance traveled by the object) because they have the same apex angle. Use the relationship of similar triangles to write an equation that connects the sides and the bases of the two triangles.A certain corner of a room is selected as the origin of a rectangular coordinate system. A fly is crawling on an adjacent wall at a point having coordinates (2.9, 2.7), where the units are meters. Express the location of the fly in polar coordinates. Looking for; radius in meters and theta is equal to what degree?
- Consider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. 1. Show that the gravitational potential 4(2) set up by that disc is given by p(2) = 2mGg | dr'; make sure to explain where the factor 27 comes from, and where the factor r' in the integrand comes from. 2. Evaluate this integral. 3. Approximate p(z), both for 0 R (i.e., for points very far away). You will need the following Taylor approximation: VI+x=1++O(x²), applied in different ways.Consider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. the gravitational potential p(2) set up by that disc is given by dr'; ()² + z² sp(2) = 27GgAn ant crawls on the surface of a ball of radius b in such a manner that the ant's motion is given in spherical coordinates by the equations r = b; phi = omega*t; theta = pi/2 * [1 + 1/4 * cos(400t)] Find the speed of the ant as a function of the time t. What sort of path is represented? by the above equations ?
- Two plane polar coordinates have the coordinates (2.2,48.3 °) and (4.6,10.50 ), calculate the distance between.Verify Kepler’s Laws of Planetary Motion. Assume that each planet moves in an orbit given by the vectorvalued function r. Let r = || r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet.