April 22, Week 14 Today: Chapter 13, Gravity Exam #4, Next Friday, - - PowerPoint PPT Presentation

Newton’s Gravity April 22, 2013 - p. 1/11

April 22, Week 14

Today: Chapter 13, Gravity Exam #4, Next Friday, April 26 Practice Exam on Website. Review Sessions: Thursday, April 25, 5:15PM, 114 Regener Hall

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass.

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by:

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 M1 - Mass of first object M2 - Mass of second object

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Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 M1 - Mass of first object M2 - Mass of second object Fg = M1M2

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 r M1 - Mass of first object M2 - Mass of second object Fg = M1M2 r - separation distance, center-to-center for spherical objects

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 r M1 - Mass of first object M2 - Mass of second object Fg = M1M2 r2 r - separation distance, center-to-center for spherical objects

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 r M1 - Mass of first object M2 - Mass of second object Fg = M1M2 r2 r - separation distance, center-to-center for spherical objects Inverse square law

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 r M1 - Mass of first object M2 - Mass of second object Fg = GM1M2 r2 r - separation distance, center-to-center for spherical objects Inverse square law

Newton’s Gravity April 22, 2013 - p. 2/11

Newton’s Law of Gravitation

Newton’s Law of Gravitation - Every object with mass exerts a gravitational force on every other object with mass. The magnitude of this force is given by: M1 M2 r M1 - Mass of first object M2 - Mass of second object Fg = GM1M2 r2 r - separation distance, center-to-center for spherical objects Inverse square law Universal Gravitational Constant: G = 6.67 × 10−11 N · m2/kg2

Newton’s Gravity April 22, 2013 - p. 3/11

The Negative Sign

For distances larger than a planet’s radius: M1 r M2 Ug = −GM1M2 r

Newton’s Gravity April 22, 2013 - p. 3/11

The Negative Sign

For distances larger than a planet’s radius: M1 r M2 Ug = −GM1M2 r

This equation comes with a built in zero.

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The Negative Sign

For distances larger than a planet’s radius: M1 r M2 Ug = −GM1M2 r

This equation comes with a built in zero. As r → ∞, Ug = 0

Newton’s Gravity April 22, 2013 - p. 3/11

The Negative Sign

For distances larger than a planet’s radius: M1 r M2 Ug = −GM1M2 r

This equation comes with a built in zero. As r → ∞, Ug = 0 When M1 and M2 are infinitely far apart Ug = 0. When objects move in the direc- tion of the force acting on them their poten- tial energy decreases ⇒ as M2 gets closer to M1, its potential energy decreases from zero ⇒ a negative amount.

Newton’s Gravity April 22, 2013 - p. 3/11

The Negative Sign

For distances larger than a planet’s radius: M1 r M2 Ug = −GM1M2 r

This equation comes with a built in zero. As r → ∞, Ug = 0 When M1 and M2 are infinitely far apart Ug = 0. When objects move in the direc- tion of the force acting on them their poten- tial energy decreases ⇒ as M2 gets closer to M1, its potential energy decreases from zero ⇒ a negative amount.

− → F

g

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Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity.

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Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞ v1 = ves =? r1 v2 = r2

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞ v1 = ves =? r1 = R (planet’s radius) v2 = r2

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞ v1 = ves =? r1 = R (planet’s radius) v2 = 0 (barely makes it) r2

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞ v1 = ves =? r1 = R (planet’s radius) v2 = 0 (barely makes it) r2 → ∞

Newton’s Gravity April 22, 2013 - p. 4/11

Escape Speed

When gravity is the only force doing work on a rocket with mass M near a planet, MP : 1 2Mv2

1 − GMP M

r1 = 1 2Mv2

2 − GMP M

r2 Escape speed - The initial speed needed by a rocket in order to barely escape from a planet’s gravity. To escape a planet’s gravity ⇒ Ug = 0 ⇒ r2 → ∞ v1 = ves =? r1 = R (planet’s radius) v2 = 0 (barely makes it) r2 → ∞ ves =

• 2GMP

R

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2.

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m (c) ∆y = −2.5 m

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m (c) ∆y = −2.5 m (d) ∆y = −5 m

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m (c) ∆y = −2.5 m (d) ∆y = −5 m (e) ∆y = −10 m

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m (c) ∆y = −2.5 m (d) ∆y = −5 m (e) ∆y = −10 m

Newton’s Gravity April 22, 2013 - p. 5/11

Review Question!

A projectile is fired horizontally on earth. (At a small height.) How far does it fall during 1 s. For simplicity use g = 10 m/s2. (a) ∆y = 0 (b) ∆y = −1 m (c) ∆y = −2.5 m (d) ∆y = −5 m (e) ∆y = −10 m Projectile: y = y0 + voyt − 1 2gt2 ⇒ ∆y = 0 − 1 2(10 m/s2)(1 s)2

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A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground?

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A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s (b) 2 s

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s (b) 2 s (c) 5 s

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s (b) 2 s (c) 5 s (d) 10 s

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s (b) 2 s (c) 5 s (d) 10 s (e) The projectile never hits the ground.

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? (a) 1 s (b) 2 s (c) 5 s (d) 10 s (e) The projectile never hits the ground.

Newton’s Gravity April 22, 2013 - p. 6/11

A New Question

The earth is not flat! It has a curvature of roughly 8000 m to 5 m (horizontal to vertical). A projectile is fired horizontally at 8000 m/s, 20 m above the ground on earth. How long does it take it to hit the ground? Projectile: x = x0 + voxt ⇒ x = (8000 m/s)(1 s) = 8000 m

The projectile follows the curvature of the earth! Every second, the projectile goes 8000 m horizontal and drops 5 m vertical. It has be- come a satellite! Satellite - Any projectile with sufficient horizontal velocity to “miss" the ground.

(e) The projectile never hits the ground.

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Orbits

Orbits come in two types:

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Orbits

Orbits come in two types: (1) Closed Orbits - Satellite returns to its starting point.

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Orbits

Orbits come in two types: (1) Closed Orbits - Satellite returns to its starting point. (2) Open Orbits - Satellite escapes to infinity.

Newton’s Gravity April 22, 2013 - p. 7/11

Orbits

Orbits come in two types: (1) Closed Orbits - Satellite returns to its starting point. (2) Open Orbits - Satellite escapes to infinity. Newton showed that when gravity is the only force doing work, the only allowed closed orbits are circular or elliptical in shape. While the only open orbits are parabolic or hyperbolic.

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Orbits II

The initial velocity of the satellite determines whether the orbit is open or closed.

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Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

2 = M2−

→ a

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Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a

Newton’s Gravity April 22, 2013 - p. 9/11

Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

2 = M2−

→ a

Newton’s Gravity April 22, 2013 - p. 9/11

Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a

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Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a ⇒ Fg = M2arad

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Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a ⇒ Fg = M2arad ⇒ Fg = M2 v2 r

Newton’s Gravity April 22, 2013 - p. 9/11

Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a ⇒ Fg = M2arad ⇒ Fg = M2 v2 r GM1M2 r2 = M2 v2 r

Newton’s Gravity April 22, 2013 - p. 9/11

Circular Orbits

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

− → F

2 = M2−

→ a ⇒ Fg = M2arad ⇒ Fg = M2 v2 r GM1M2 r2 = M2 v2 r v =

• GM1

r

Newton’s Gravity April 22, 2013 - p. 10/11

Circular Orbits II

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

Speed: v =

• GM1

r

Newton’s Gravity April 22, 2013 - p. 10/11

Circular Orbits II

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

Speed: v =

• GM1

r Period, T - Time for one revolution

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Circular Orbits II

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

Speed: v =

• GM1

r Period, T - Time for one revolution Constant Speed ⇒ v = 2πr T

Newton’s Gravity April 22, 2013 - p. 10/11

Circular Orbits II

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

Speed: v =

• GM1

r Period, T - Time for one revolution Constant Speed ⇒ v = 2πr T Period: T = 2πr3/2 √GM1

Newton’s Gravity April 22, 2013 - p. 10/11

Circular Orbits II

In circular orbit, gravity creates the centripetal acceleration. M1 r M2 − → F

g

Speed: v =

• GM1

r Period, T - Time for one revolution Constant Speed ⇒ v = 2πr T Period: T = 2πr3/2 √GM1 Energy: E = −GM1M2 2r

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Kepler’s Laws

Before Newton, all astronomical work had been observational. Using the data of Danish astronomer Tycho Brahe (1546-1601), the German mathematician Johannes Kepler (1571-1630) was able to deduce (but not explain), three statements about planetary motion.

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Kepler’s Laws

Before Newton, all astronomical work had been observational. Using the data of Danish astronomer Tycho Brahe (1546-1601), the German mathematician Johannes Kepler (1571-1630) was able to deduce (but not explain), three statements about planetary motion. Kepler’s Laws: 1: Each planet’s orbit traces out the shape of an ellipse with the sun located at one focus. 2: The imaginary line from the sun to a planet sweeps

• ut equal areas in equal times.

3: The period of the planet’s motion is proportional to the orbit’s semi-major axis to the 3

2 power.