Gravitational Field and Acceleration Due to Gravity The gravitational field, g, is a vector quantity. The gravitational field depends upon 2 things: The mass of the planet creating the field. The distance from the center of the planet. On Earth, the gravitational field is 10 N/kg. This is the “g” in W = mg. This should not be confused with the other “g”, which is acceleration due to gravity. Aren’t they the same thing? Yes? No? Maybe? Here’s the physics. First, g = 10 N/kg only close to the Earth’s surface. As you move further away from the Earth, the gravitational field is less than 10 because of the inverse square law. So, let’s isolate our study to close to the surface where g = 10 N/kg. Let’s pretend we have a mass m. The FORCE of gravity on the object is Fg = mg. You pick the object up and drop it, gravity is the only force acting on it, so F = mg. That means by Newton’s second law that mg = ma, or by canceling the m on both sides, a = g. this means the acceleration due to gravity close to the surface of the Earth is equivalent to g. Not clear? How about this. As you lift an object further from the center of the Earth, the gravitational force gets less. Again, the inverse square law. So, the acceleration must also be less. 푀푚 ΣF = ma = G 푟2 푀 a = G 푟2 This means that the acceleration of any object towards the surface of a planet is equal to the gravitational field at the location of the object. So as the object falls to the Earth, its acceleration increases. If that isn’t mind boggling enough, the object applies the same force to the Earth that the Earth applies to it. So,… 푀푚 ΣF = Ma = G 푟2 푚 a = G 푟2 The Earth accelerates to towards the object at the same time that the object accelerates towards the Earth. Gravitational and Inertial Mass Gravitational mass indicates how an object responds to a gravitational field. Inertial mass indicates how an object accelerates in response to a net force. In every experiment ever conducted, an object’s gravitational mass is equal to its inertial mass. Classwork 1. Compute g for the surface of Mercury. Its radius is 2.4 x106 m and its mass is 3.3 x 1023 kg. 2. An object experiences an acceleration, b, on a planet with mass M and radius R. The object is moved to a second planet where the mass is 5M and radius is 2R. What is the acceleration the object experiences on this planet in terms of b? 3. An object experiences an acceleration of a on a planet with mass M and radius R. The object is moved to a distance of 2R above the surface of the planet. What is the acceleration of the object at this point? 4. NASA sends a 20-kg rover to Planet X, which has twice the mass and twice the diameter of Earth. What is the weight of the rover on Planet X? 5. The Earth has a mass of 6.0 × 1024 kg. The sun has a mass of 2.0 × 1030 kg. The Earth orbits the sun in a circle with a radius of 1.5 × 1011 m. Assume all outside gravitational forces are negligible. What is the gravitational force the Earth applies to the sun? 6. Two spherical objects have masses of 3.1 x 105 kg and 6.5 x 103 kg. The gravitational attraction between them is 65 N. How far apart are their centers? 7. Two spherical objects have equal masses and experience a gravitational force of 25 N towards one another. Their centers are 36cm apart. Determine each of their masses. 8. An 800.0 kg mass and a 400.0 kg mass are placed 5.0 m apart. A 200.0 kg mass is placed somewhere on the direct line between these two mass so that the net gravitational force on it is zero. How far is this point from the 400.0 kg mass? .