Gravitational Force Between Two Bodies
Universal Attraction: Every single object, no matter how big or small, pulls on every other object in the universe through a force called gravity. This force is always there, and it acts across even very large distances.
Massive Objects Noticeable: The force of gravity is always present, but we usually only notice it when at least one of the objects involved is very massive—like a planet, moon, or star—because more mass means more pull.
Force Formula: The gravitational force between two objects depends on how heavy (massive) they are and how far apart they are. If the masses are big, the force is stronger. If the distance is large, the force is weaker.
Mathematical Expression: We can calculate this force using the formula F = G × (m₁m₂ / r²). In this formula, F is the gravitational force, m₁ and m₂ are the masses of the two objects, r is the distance between their centers, and G is a special number called the gravitational constant.
Gravitational Constant: G is a fixed number that doesn’t change. It is equal to about 6.67 × 10⁻¹¹ Newton metre² per kilogram². This very small number shows how weak gravity is unless the masses are very large.
Attractive Force: Gravity is always attractive, which means it always pulls objects toward each other. It never pushes them away.
Mass Increases Force: If either of the objects has more mass, the gravitational force becomes stronger. So, heavier objects create stronger pulls.
Distance Reduces Force: If you move two objects farther apart, the gravitational pull between them becomes weaker. The farther they are, the less they pull on each other.
Effect of Mass and Distance on Gravitational Force
Mass-Proportional Effect: When the distance between two objects stays the same, increasing either object’s mass increases the gravitational force. For example, doubling the mass of one object makes the force twice as strong.
Examples of Mass Effect: If both masses are doubled, the force becomes four times stronger. If one mass is tripled, the force becomes three times stronger.
Distance-Inverse Square Law: The gravitational force gets weaker much faster as the distance increases. If you double the distance, the force becomes one-fourth as strong. If you triple the distance, the force becomes one-ninth as strong.
Examples of Distance Effect: For example, if two objects are 10 meters apart, and we increase the distance to 20 meters, the gravitational force is only one-fourth as strong as before.
Solving Problems
Using the Formula: To find the gravitational force between two objects, you just plug the values for mass and distance into the formula F = G × (m₁m₂ / r²). This helps us solve real-world problems involving gravity.
Practical Example: You can use this formula to find the gravitational pull between two people or between a person and the Earth, as long as you know the masses and the distance between them.
Body on Earth's Surface
Weight as Gravity: The reason we feel weight is because Earth’s gravity is always pulling us down toward the center of the planet. When you stand on the ground, this invisible pulling force is what you feel as your weight. So, your weight is simply the force of gravity acting on the mass of your body, keeping you grounded.
Cause of Acceleration: Gravity is the force that causes things to fall when you let go of them. It pulls every object toward Earth’s center, which causes them to speed up or accelerate as they fall. The longer they fall, the faster they go, unless something like air slows them down.
Gravity Symbol: Scientists use the lowercase letter ‘g’ to represent the acceleration caused by gravity. This symbol tells us how quickly the speed of a falling object increases. On Earth, the value of g tells us that every second, a falling object’s speed increases by about 9.81 meters per second.
Weight Equation: The formula F = mg is a simple way to calculate an object’s weight. In this formula, F is the weight or gravitational force, m is the object’s mass, and g is the acceleration due to gravity (about 9.81 m/s² on Earth). This tells us that the more mass something has, the more it weighs.
Alternate Expression for Weight: Another way to figure out weight comes from Newton’s Law of Universal Gravitation. It says that weight can also be calculated using F = G × (Mm / R²), where G is the gravitational constant, M is Earth’s mass, m is the object’s mass, and R is the radius of Earth. This formula explains how distance and mass affect gravitational force.
Derivation of g: If you compare the two weight formulas, mg = G × (Mm / R²), you can cancel out the object’s mass, m, on both sides. This gives a new formula: g = G × (M / R²). This equation shows how the value of g (gravitational acceleration) depends on Earth’s mass and how far you are from its center.
Calculating g: To find the exact value of g using the formula g = G × (M / R²), you need to plug in the correct numbers. Use G = 6.67 × 10⁻¹¹ N·m²/kg², M = 5.97 × 10²⁴ kg (Earth’s mass), and R = 6.37 × 10⁶ m (Earth’s radius). When you do the math, you get g ≈ 9.81 m/s².
Standard Value of g: The standard value of gravitational acceleration on Earth’s surface is about 9.81 meters per second squared. This means if an object is falling freely, every second it gets 9.81 m/s faster until it hits the ground or something stops it.
g Variation by Shape: Earth isn’t a perfect sphere—it’s slightly squashed at the poles and bulging at the equator. Because people at the poles are a bit closer to Earth’s center, gravity is slightly stronger there than it is at the equator.
Altitude Effect on g: The higher you go above Earth’s surface, the weaker gravity becomes. For example, if you go up a tall mountain or ride in an airplane, you’re farther from the center of the Earth, and gravity pulls on you a little less.
Depth Effect on g: If you dig deep into Earth or go down a mine, gravity becomes slightly weaker too. Even though you’re getting closer to the center, some of Earth’s mass is now above you, so there’s less gravitational pull from below.
Relationship Between g and G
Relationship Between g and G: The connection between g and G is shown in the formula g = G × (M / R²). This formula comes from setting Newton’s second law of motion (F = mg) equal to Newton’s law of gravity (F = G × (Mm / R²)), and simplifying by canceling the mass of the object.
g is Mass-Independent: One important fact about gravity is that it doesn’t care how heavy or light something is. In a place with no air, like outer space, a bowling ball and a feather would fall at the exact same rate because g is the same for all objects.
g Depends on Earth Only: The value of g stays the same for all objects near Earth’s surface because it depends only on Earth’s mass and radius—not on the object’s weight or size.
Importance of Knowing g in the Solar System
Importance of Knowing g in the Solar System: Scientists use the formula g = GM / r² to figure out how strong gravity is on other planets, moons, or stars. Knowing g on those worlds helps astronauts plan space missions and understand what it would feel like to stand or move there.
Weight on Other Planets: Because different planets have different masses and sizes, the gravity—and therefore your weight—changes from planet to planet. Even though your mass doesn’t change, the force of gravity pulling on you does. That’s why you weigh less on the Moon than on Earth.
Mercury Example: Let’s say your weight on Earth is 1000 newtons. If you went to Mercury, your weight would drop to around 380 newtons. That’s because Mercury’s gravity is much weaker—it’s only about 19/50 of Earth’s gravity.
Space Exploration and Gravity
Space Exploration and Gravity: Rockets and spacecraft don’t just use fuel to move—they also use gravity from planets and moons to change direction and speed. This method is called a gravity assist and it helps save fuel during long space missions.
Satellite Observations: Satellites that orbit Earth are held in place by gravity. These satellites can take photos of Earth, track weather patterns, or look out into space to help scientists study other planets and stars.
Safe Landings: Gravity plays a key role in helping spacecraft land safely on planets or moons. As they descend, gravity pulls them down, and engineers design systems to make sure that pull doesn’t cause a crash but a smooth landing.
Telecommunications and Satellites
Telecommunications and Satellites: Gravity keeps communication satellites in orbit. These satellites help us make phone calls, use GPS, browse the internet, and watch TV by sending and receiving signals across long distances.
Weather Forecasting: Satellites in orbit can observe Earth’s weather from above. They send back images of clouds, storms, and temperature changes to help meteorologists make accurate weather predictions.
Medical Research in Space
Zero Gravity Experiments: In space, gravity is almost zero. This condition, called microgravity, allows scientists to study how bones, muscles, and body systems work without the constant downward pull we feel on Earth.
Human Growth and Gravity
Human Growth and Gravity: Gravity affects how blood moves in your body, but it doesn’t change the size of your lungs. No matter where you are, your lungs stay the same size—but gravity can affect how well they function.
Brittle Bones in Low g: In low-gravity environments like space, astronauts’ bones don’t carry weight like they do on Earth. Over time, this can cause the bones, especially in the hips and spine, to become weaker or brittle.
Circulatory Effects: In space, the heart doesn’t have to work as hard to move blood throughout the body. This can cause blood pressure to drop and circulation to slow down, which is why astronauts must be monitored closely.
Centripetal Force in Motion
Centripetal Force in Motion: When something moves in a circle, like a ball on a string or a car turning, it needs a force to keep pulling it toward the center. This inward-pulling force is called centripetal force and keeps the object moving in its circular path.
Centripetal Acceleration: As an object moves around in a circle, its direction keeps changing, which means it is accelerating. This kind of acceleration is called centripetal acceleration and it always points toward the center of the circle. You can calculate it with ac = v² / r.
Relating Force and Acceleration: The formula Fc = mac tells us that centripetal force equals mass times centripetal acceleration. This shows how an object’s mass and how fast it’s turning affects how much force is needed to keep it in a circle.
Centripetal Force Formula: When we substitute the formula for centripetal acceleration (ac = v² / r) into Fc = mac, we get Fc = mv² / r. This version shows that the required force depends on the object’s speed, mass, and how tight the circle is (its radius).
Gravity as Centripetal Force: For planets and satellites, gravity acts as the centripetal force that keeps them in orbit. Without gravity pulling them toward the center, they would fly off into space in a straight line.
Maintaining Orbit: Gravity curves the path of planets and satellites so they travel around in circles or ellipses. This keeps them from flying away and helps them stay in their proper orbits.
No Force, No Orbit: If there were no force pulling an object toward the center, it wouldn’t go in a circle—it would just shoot off in a straight line. That’s why centripetal force is essential for orbiting motion.
Satellite Free Fall: Satellites orbit Earth by falling toward it, but they’re also moving forward fast enough that they keep missing it. This creates a stable orbit where they’re constantly falling but never hitting the ground.
Determining the Mass of Earth and Sun
Earth’s Mass Formula: We can figure out Earth’s mass using a rearranged formula from earlier: M = gR² / G. This comes from setting the formula for weight equal to the gravitational force and solving for M.
Earth’s Mass Value: When we plug in values for g, R, and G into M = gR² / G, we find that Earth’s mass is about 5.97 × 10²⁴ kilograms. That’s a number so big, it’s hard to imagine—but it tells us just how huge and heavy our planet is.
Sun’s Mass Basis: The reason Earth moves in a circle around the Sun is because of the Sun’s gravity. That gravitational force acts like the centripetal force that keeps Earth in orbit.
Equating Forces for Sun: To figure out the Sun’s mass, we can set the Sun’s gravitational force equal to the centripetal force needed to keep Earth moving in a circle. This gives us the formula GM / r = v².
Using Orbital Velocity: If we know how far Earth is from the Sun and how long it takes to orbit (1 year), we can use the formula M = 4π²r³ / GT² to calculate the Sun’s mass. This comes from plugging orbital speed into the force equation.
Sun’s Mass Value: Using real values for Earth’s orbit, we can calculate the Sun’s mass to be about 2.01 × 10³⁰ kilograms. That’s more than 300,000 times as massive as Earth—making the Sun the heavyweight of our solar system.
Alternative Earth Mass Method: Another way to calculate Earth’s mass is by observing the Moon’s motion. By comparing the gravitational force pulling on the Moon to the centripetal force keeping it in orbit, we can work backward to figure out Earth’s mass.