Universal Gravitation Honors Physics First! Mr. Fowler

Objectives: Contrast Mass and Weight. Compute Mass or Weight given the other. Describe Newton’s Law of Universal Gravitation. Use Newton’s Law of Universal Gravitation to solve problems involving gravitational forces. Explain gravitational force and how it affects the motion of an object. Describe the motion of an object in Free Fall, and solve simple problems.

Weight vs. Mass Compare the weight and mass of an object, such as a brick, on the earth and on the moon. –The object’s mass is the same on both the Moon and Earth, but its weight is less on the Moon. Weight depends on mass AND the strength of gravity at a certain place. –An object’s weight is different on other planets, at high altitude, etc.

Weight vs. Mass Mass does NOT depend on gravity. –Mass changes only when matter is added to or taken away from an object. Weight is the Force that gravity exerts. F g = ma g a g = 9.8 m/s 2 on earth’s surface.

Weight Sample Problem Sample problem: What is the weight on earth’s surface (in Newtons) of a 20-kg wolverine? F g = ma g = (20 kg)(9.8 m/s 2 ) = 196 N

Weight Practice Problem What is the weight on earth’s surface (in Newtons) of a 50-kg Ostrich? F g = ma g = (50 kg)(9.8 m/s 2 ) = 490 N

Weight in Pounds On earth’s surface, a mass of 1 kg has a weight of 2.2 lbs. What is the 20-kg wolverine’s weight in pounds? F g = ma g = (20 kg)(2.2 lbs/kg) = 44 lbs What is the Ostrich’s weight in pounds? F g = ma g = (50 kg)(2.2 lbs/kg) = 110 lbs

Weight Problems What is the weight in Newtons and pounds of a 6-kg bowling ball? F g = ma g = (6 kg)(9.8 m/s 2 ) = 58.8 N F g = ma g = (6 kg)(2.2 lbs/kg) = 13.2 lbs What is the mass of a 75-lb boy? m = F g /a g = (75 lbs)/(2.2 lbs/kg) = 34 kg What is the mass of a 650-N dog? m = F g /a g = (650 N)/(9.8 m/s 2 ) = 66.3 kg

Weight Problems These calculations are similar to a unit conversion, since earth’s gravity is consistent at its surface. 1kg ≈ 9.8 N 1 kg ≈ 2.2 lbs They are not truly a unit conversion, however, since they relate different quantities, mass and weight.

Universal Gravitation On what factors does the strength of gravity depend? –Mass and distance.

Universal Gravitation Newton's Law of Universal Gravitation Where: F g is the force of gravitational attraction between any two objects. G is a universal gravitational constant. G = 6.67 X Nm 2 /kg 2 m A & m B are the masses of objects A & B. d is the distance between the centers of the two objects.

Sample Problem: Find the gravitational attraction between George (mass 85 kg) and Sally (mass 65 kg), standing 10-m apart at a party. How does the attraction change as they get closer together? Using Newton’s Law of Universal Gravitation: As they get closer together, the attraction becomes stronger.

Two bowling balls each have a mass of 6.8 kg. They are located next to each other with their centers m apart. What gravitational force do they exert on each other? Using Newton’s Law of Universal Gravitation:

Two rocks have their centers 2.0 m apart. One rock has a mass of 8.0 kg; the other has a mass of 6.0 kg. What is the gravitational force between them? Using Newton’s Law of Universal Gravitation:

Do heavier objects fall faster? The force of gravity acting on an object near Earth’s surface is proportional to the mass of the object. Why does a heavy object not fall faster than a light object? –It’s true that the force of gravity on an object is greater with greater mass. –But resistance to accelerate is also greater with greater mass. (That’s Inertia!) –These two effects offset each other, such that an object’s acceleration caused by the force of gravity equals its weight divided by its mass. a g = F g /m

Do heavier objects fall faster? In most real situations, where objects are falling through air, there is another force besides gravity. –Air resistance is a force that opposes the motion of an object, partially balancing gravity for a falling object. –Air resistance depends on many factors, including speed, shape, surface characteristics, and area. Some objects build air resistance quickly, while others take longer for air resistance to matter. Leaf vs Apple Flat paper vs Crumpled paper Parachute open vs Parachute closed. –Terminal velocity occurs when the air resistance force totally balances gravity. The object stops accelerating.

Reconciling F g = ma g with F g = Gm A m B /d 2 First, some terms and substitutions: F g for both equations represents the weight of an object; the force of attraction between the object and Earth. m = m A (mass of the object) m B = m E (mass of Earth = 5.97 x kg) d = r E (radius of Earth = 6.38 x 10 6 m) so: and: Cancel m:

Do heavier objects fall faster? The force of gravity acting on an object near Earth’s surface is proportional to the mass of the object. Why does a heavy object not fall faster than a light object? –It’s true that the force of gravity on an object is greater with greater mass. –But resistance to accelerate is also greater with greater mass. (That’s Inertia!) –These two effects offset each other, such that an object’s acceleration caused by the force of gravity equals its weight divided by its mass. a g = F g /m

Do heavier objects fall faster? In most real situations, where objects are falling through air, there is another force besides gravity. –Air resistance is a force that opposes the motion of an object, partially balancing gravity for a falling object. –Air resistance depends on many factors, including speed, shape, surface characteristics, and area. Some objects build air resistance quickly, while others take longer for air resistance to matter. Leaf vs Apple Flat paper vs Crumpled paper Parachute open vs Parachute closed. –Terminal velocity occurs when the air resistance force totally balances gravity. The object stops accelerating.

Reconciling F g = ma g with F g = Gm A m B /d 2 First, some terms and substitutions: F g for both equations represents the weight of an object; the force of attraction between the object and Earth. m = m A (mass of the object) m B = m E (mass of Earth = 5.97 x kg) d = r E (radius of Earth = 6.38 x 10 6 m) so: and: Cancel m:

1. What is the gravitational force between a 2200-kg SUV and a 22,000-kg tractor-trailer, just 3 meters apart on the highway? Using Newton’s Law of Universal Gravitation:

2. What is the gravitational force between a 4000-kg elephant and his 3000-kg mate when they stand 2.5-m apart? Using Newton’s Law of Universal Gravitation:

3. Compare calculation of a man’s weight on earth using F g = ma g and Newton’s Law of Gravitation. Using Newton’s Law of Universal Gravitation: Using F g = ma g = (84 kg)(9.8 m/s 2 ) = 823 N Both calculations are about the same. Note that if F g = ma g, and Then

4. What would happen to the value of a g on Earth’s surface if: a.Earth began to shrink, but its mass remained the same? The value of a g would increase. b.If earth were twice as massive but remained the same size a g would double. c.If earth’s radius doubled, but its mass remained the same? a g would be 1/4 th its current value.

5. What is the value of Earth’s a g for a satellite orbiting at 200 km above Earth’s surface?

6. Jupiter has about 300 times the mass of Earth and about 10 times Earth’s radius. Estimate the size of a g on the surface of Jupiter. Check the formula above. If we multiply m E by 300 and r E by 10, a g triples. So Jupiter’s a g would be about 29.4 m/s 2.

7. As you walk past a skyscraper in a city, it exerts a gravitational force pulling you toward it. Why don’t you feel this force? The force is much too small for you to feel it, compared to other forces acting on you. 8.The mass of an object is usually measured with a scale, relying on the force of gravity between the object and the scale. What is the mass of an object that weighs 24.5 N? F g = ma g  m = F g /a g = (24.5 N)/(9.8 m/s 2 ) = 2.5 kg

9. What is the weight in Newtons and In Pounds of : a.a 65-kg man? 637 N143 lbs b.a 4.5-kg cat? 44.1 N9.9 lbs c.a 2600-kg truck? 25,480 N5720 lbs