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 . Use Proportional Reasoning (the concept of proportions when analyzing and solving a mathematical situation).

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.

TEDed Talks about Gravity https://www.youtube.com/watch?v=lY3XV_GGV0M

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 and 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 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

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 (6371 km). 𝐹 𝑔 =𝐺 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑅 𝐸𝑎𝑟𝑡ℎ 2 ×𝑚 𝑔=𝐺 𝑀 𝐸𝑎𝑟𝑡ℎ 𝑅 𝐸𝑎𝑟𝑡ℎ 2

Derivation of g 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.

Weightlessness & Inertial Mass Humans experience their own body weight as a result of a normal force applied to a person by the surface of a supporting object on which the person is standing or sitting. In the absence of this force, a person would be in free-fall, and would experience weightlessness. Weightlessness Weight

Regular Physics (Period 2) Summary Summarize the concepts of gravity and gravitational field strength in your own words. What is the equation for the gravitational force between any two bodies? Outline the process for proportional reasoning. What does it mean for something to be “weightless”? Does gravity exist in space?

IB Example Proportional Reasoning 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.

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

IB Objectives State the definitions of gravitational potential energy, 𝐸 𝑃 =−𝐺 𝑀 1 𝑀 2 𝑟 , and gravitational potential, 𝑉=−𝐺 𝑀 𝑟 ; Understand that the work done as a mass m is moved across 2 points with gravitational potential difference ΔV is 𝑊=𝑚∆𝑉; Understand the meaning of escape velocity, and solve related problems using the equation for escape speed from a body of mass M and radius R; 𝑣 𝑒𝑠𝑐 = 2𝐺𝑀 𝑅 ; Solve problems for orbital motion using the equation for orbital speed at a distance r from a body of mass M: 𝑣= 𝐺𝑀 𝑟

Gravitational Potential Energy Consider masses M and m placed in space a distance R from each other. The two masses have gravitational potential energy, which is stored in their gravitational field. This energy is there because work had to be done to move one of the masses from infinity to the position near the mass. 𝐸 𝑃 =−𝐺 𝑀 1 𝑀 2 𝑅

Case: A Satellite orbiting Earth 𝐸 𝑇𝑜𝑡𝑎𝑙 = 𝐸 𝑘 + 𝐸 𝑝 = 1 2 𝑚 𝑣 2 −𝐺 𝑀𝑚 𝑟 Simplified by the universal law of gravitation and newton’s 2nd law to 𝐺 𝑀𝑚 𝑟 2 =𝑚 𝑣 2 𝑟 → 𝑣 2 = 𝐺𝑀 𝑟 → 𝐸 𝐾 = −𝐺𝑀𝑚 2𝑟 This shows E= −𝐺𝑀𝑚 2𝑟 or E=− 1 2 𝑚 𝑣 2

Gravitational Potential Related to the concept of gravitational potential energy is that of gravitational potential, V The gravitational potential is a field because it is defined at every point in space, but unlike the field it is a scalar quantity (lacking direction) The gravitational potential at a point P in the gravitational field is the work done per unit mass in bringing a small point mass m from infinity to point P. 𝑉= 𝑊 𝑚 =−𝐺 𝑀 𝑟 Units: J kg-1

Gravitational Potential If we know that a mass, or arrangement of masses, produces a gravitational potential V at some point in space, then putting a mass m at that point means that gravitational potential energy of the mass will be 𝐸 𝑝 =𝑚𝑉. If a mass is positioned at a point in a gravitational field where the gravitational potential is V1 and is moved to another point of gravitational potential V2, then the work done on the mass is 𝑊=𝑚 𝑉 1 − 𝑉 2 =𝑚∆𝑉

Example The graph shows the variation of the gravitational potential due to a planet with distance, r. Using the graph, estimate: The gravitational potential energy of an 800 kg spacecraft that is at rest on the surface of the planet. The work done to move this spacecraft from the surface of the planet to a distance of four planet radii from the surface of the planet

Total energy must be zero or positive Escape velocity How fast does it have to go to not come back down? The total energy of a mass m moving near a large stationary mass M is 𝐸= 1 2 𝑚 𝑣 2 −𝐺 𝐺𝑀𝑚 𝑟 ; where v is the speed of m when it is a distance r from M. (if M is also free to move, then you need to include a term 1 2 𝑀 𝑢 2 ; where u is the speed of M.) The only force acting on m is the gravitational attraction of M. Suppose that m is launched with a speed v0 from M. Will m escape from the pull of M and move very far away from it? 1 2 𝑚 𝑣 0 2 −G 𝑀𝑚 𝑅 = 1 2 𝑚 𝑣 ∞ 2 Total energy must be zero or positive

Escape velocity E>0: mass escapes and never returns E<0: mass moves out a certain distance, but returns – trapped E=0: the critical case separating the other two – mass barely escapes Which of these would be good for a space shuttle?

Not dependent on the mass of the object Escape velocity How fast something must go in order to escape the gravitational pull of an object. Smallest v for which v∞ = 0 Kinetic energy = gravitational potential energy 1 2 𝑚 𝑣 2 =𝑚𝑔ℎ 𝑣= 2𝑔ℎ = 2𝐺𝑀 𝑟 Not dependent on the mass of the object Does not account for frictional forces (air resistance) or other external forces.

Example – the Swarzchild Radius What must the radius of a star of mass M be such that the escape velocity from the star is equal to the speed of light, c?

Example Compute the Swarzchild radius of the earth and the sun.

IB Summary Summarize the concepts of gravity and gravitational field strength in your own words. What is the equation for the gravitational force between any two bodies? Outline the process for proportional reasoning. What does it mean for something to be “weightless”? Does gravity exist in space? Compare gravitational potential and gravitational potential energy. Include equations. What is escape velocity? Include the equation.