Universal Gravitation Gravity sucks!

Objectives Appreciate that there is an attractive force between any two point masses 𝐹=𝐺 𝑀 1 𝑀 2 𝑟 2 ; State the definition of gravitational field strength, 𝑔=𝐺 𝑀 𝑟 2 .

Newton’s law of Gravitation Any object falling near the surface of the Earth accelerates toward the ground at 9.8 m s-2, and thus experiences a net force in the direction of its acceleration (Weight). Similarly, a planet that revolves around the sun also experiences (centripetal) acceleration, thus a force must be acting on it. Newton’s hypothesis: same force! Gravity! Conventional weight is the gravitational force of attraction between that body and the Earth.

THE EQUATION 𝑭=𝑮 𝑴 𝟏 𝑴 𝟐 𝒓 𝟐 Where: F is the force of gravity G is the gravitational constant (6.667x10 -11 Nm2kg-2) M1 and M2 are the masses of the attracting bodies r is the separation between them Direction: along the line joining the two masses.

Limitation(s)? Applies only to point masses BUT: Masses are very small compared to separation BUT: In the case of large objects (sun & planet), the formula still applies because the separation is much larger than the radii For spherical bodies of uniform density, one can assume that the entire mass of the body is concentrated at the center – as if the body is a point mass.

Example Find the force between the Sun an the Earth. (R = 1.5x1011 m; Me = 5.98x1024 kg, Ms = 1.99x1030 kg)

Example – Proportional Reasoning If the distance between 2 bodies is doubled, what happens to the gravitational force between them?

Gravitational Field Strength The force we usually call weight (W = mg) is the gravitational attraction between the mass of the Earth (Me) and the mass of the body in question. Me is assumed to be concentrated at its center, thus the distance that goes in the formula is the radius of the Earth, Re.

Derivation Therefore, we must have that 𝐺 𝑀 𝑒 𝑚 𝑅 𝑒 2 =𝑚𝑔 𝐺 𝑀 𝑒 𝑅 𝑒 2 =𝑔 This relates the acceleration of gravity to the mass and radius of the Earth. Thus the acceleration due to gravity on Jupiter is 𝐺 𝑀 𝐽 𝑅 𝐽 2 =𝑔!

Example Find the acceleration due to gravity (the gravitational field strength) on a planet 10 times as massive as Earth and with radius 20 times as large.

Example Find the acceleration due to gravity at a height of 300 km from the surface of the Earth.

Gravitational Field Strength How another mass “knows” a mass is nearby. All masses create gravitational fields simply by existing. Other masses “feel” this as a force. Definition: the gravitational field strength at a certain point is the force per unit mass experienced by a small point mass m at that point. Units: N kg-1 Direction: the direction of the force a point mass would experience if placed at that point.

Gravitational Field Strength Radial – same for all points equidistant from the mass, directed toward the mass Also true outside a uniform spherical mass Constant gravitational field strength – flat mass

Example Two stars have the same density, but star A has double the radius of star B. Determine the ratio of the gravitational field strength at the surface of each star.

Example Show that the gravitational field strength at the surface of a planet of density ρ has a magnitude given by 𝑔= 4𝐺𝜋𝜌𝑅 3 .