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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 p(z) set up by that disc is given by rR 9(2) = 2mGo | dr'; (r)² + 2 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 < z « R (i.e., for points very close to the disc) and for z » R (i.e., for points very far away). You will need the following Taylor approximation: VI+I=1++O(z*), 2 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. 1. Show that the gravitational potential p(z) set up by that disc is given by rR 9(2) = 2mGo | dr'; (r)² + 2 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 < z « R (i.e., for points very close to the disc) and for z » R (i.e., for points very far away). You will need the following Taylor approximation: VI+I=1++O(z*), 2 applied in different ways.

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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 < z « R (i.e., for points very close to the disc) and for z > R (i.e., for points very far away). You will need the following Taylor approximation: VI+x=1++O(x²), applied in different ways.
Transcribed Image Text: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 < z « R (i.e., for points very close to the disc) and for z > R (i.e., for points very far away). You will need the following Taylor approximation: VI+x=1++O(x²), applied in different ways.
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