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Interactive Audio Lesson
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Gravitational Force Above Earth
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Let's explore how gravity changes when we move above the Earth's surface. Can anyone tell me how the distance to the center of Earth affects gravity?
I think the further you go from the Earth, the less gravity you feel!
Exactly! The force can be calculated using the formula \( F(h) = \frac{GM_E m}{(R_E + h)^2} \). What happens to \( g(h) \) as \( h \) increases?
It decreases, right?
Correct! As you go higher, gravity decreases less significantly according to our surface gravity equation \( g = \frac{GM_E}{R_E^2} \). Let's remember this pattern: the higher you go, the less you weigh β 'h' for height and 'h' for lesser gravity!
Gravitational Force Below Earth
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Now let's switch gears and discuss gravity beneath Earth's surface. Who can guess how gravitational pull changes when we dig down?
I think it decreases too!
Very close! The force due to gravity below the surface is calculated with \( F(d) = \frac{GM_S m}{(R_E - d)^2} \). How does this look compared to our earlier equation?
It seems like we're using a different radius!
Exactly! As we go deeper, we use the radius minus our depth. The acceleration becomes \( g(d) = g \left(1 - \frac{d}{R_E}\right) \). So, deeper means less gravity, and we can remember this: 'Deep Equals Decreased gravity β DED.'
Introduction & Overview
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Quick Overview
Standard
The section explains the mathematical relationship for gravitational acceleration at various heights above and below the Earth's surface. It delineates how gravity decreases at height and depth and provides equations to calculate it, emphasizing the unique characteristics of gravitational force in these contexts.
Detailed
Acceleration Due to Gravity Below and Above the Surface of Earth
This section elaborates on the concept of gravitational acceleration (g) at different positions relative to Earthβs surface. It separates the discussion into two main areas: acceleration at heights above the Earth's surface and acceleration at depths below the surface.
Gravitational Force Above Earth's Surface
When a mass m is positioned at a height h above the Earth, the gravitational force acting on it can be described by the formula:
\[ F(h) = \frac{GM_E m}{(R_E + h)^2} \]
where G is the gravitational constant and \( R_E \) is the radius of the Earth. From this force, we derive acceleration due to gravity:
\[ g(h) = \frac{F(h)}{m} = \frac{GM_E}{(R_E + h)^2} \]
This indicates that as h increases, g(h) decreases relative to the surface gravity, diminishing with height.
For small heights (\( h << R_E \)), we can approximate g(h) using a binomial expansion:
\[ g(h) \approx g - \frac{g h}{R_E} \]
where g is the gravitational acceleration at the Earth's surface.
Gravitational Force Below Earth's Surface
Conversely, at a depth d below the surface, the calculation changes. The force on a mass m at that depth can be expressed as:
\[ F(d) = \frac{GM_S m}{(R_E - d)^2} \]
where \( M_S \) is the mass of the smaller sphere of radius \( (R_E - d) \) that contributes to the gravitational pull. The total gravitational acceleration then becomes:
\[ g(d) = g \left(1 - \frac{d}{R_E}\right) \]
This depicts how gravitational acceleration diminishes linearly as one travels deeper into the Earth. Notably, gravity is maximal at the surface and decreases consistently as one moves both upward to heights and downward into depths. Thus, the behavior of gravitational acceleration is fundamental for understanding various physical phenomena related to gravity.
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Acceleration Above Earth's Surface
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Consider a point mass m at a height h above the surface of the earth as shown in Fig. 7.8(a). The radius of the earth is denoted by RE. Since this point is outside the earth,
$$F(h) = \frac{G M_E m}{(R_E + h)^2}$$.
The acceleration experienced by the point mass is $$g(h) = \frac{F(h)}{m}$$ and we get
$$g(h) = \frac{G M_E}{(R_E + h)^2}$$
This is clearly less than the value of g on the surface of earth: $$g_E = \frac{G M_E}{R_E^2}$$.
Detailed Explanation
When a mass is at a height h above the Earth's surface, we can derive the gravitational force using Newton's law of gravitation. This force diminishes with distance, illustrated with the formula for gravitational force at height h: F(h) = (G M_E m) / (R_E + h)Β², where G is the gravitational constant and M_E is Earth's mass. To determine the acceleration experienced by the mass at this height, we rearrange this equation to isolate acceleration, leading to the formula g(h) = (G M_E) / (R_E + h)Β². This shows that as we move further away from Earth's surface, the gravitational pull (and hence acceleration) decreases compared to the value at the surface.
Examples & Analogies
Imagine being on a high mountain or in a plane. The higher you go, you feel lighter. This is similar to how the pull of Earth's gravity decreases with height, similar to how the force of a magnet decreases as you move it further away.
Expanded Form for Small Heights
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For small heights h where $$h<<R_E$$, we can expand the RHS of the above equation:
$$g(h) \approx g_E \left(1 - \frac{h}{R_E}\right)$$.
Detailed Explanation
This chunk introduces a more refined approximation for gravitational acceleration at minor heights above the Earth's surface. By using a binomial expansion, we can simplify the equation to g(h) β g_E (1 - (h / R_E)). This means that the gravitational acceleration decreases linearly with height as you go higher, enabling us to begin to predict how much lighter an object becomes as we ascend.
Examples & Analogies
If you've ever climbed a hill, you may have noticed that it feels easier to lift a light backpack at the top than at the bottom. This feeling represents how gravity lessens with height. Here, the backpack becomes slightly lighter as you go higher because the force of gravity diminishes.
Acceleration Below Earth's Surface
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Now, consider a point mass m at a depth d below the surface of the earth. For the smaller sphere of radius (R_E - d), the point mass is outside it and the force on m is as if the mass of this smaller sphere is concentrated at the center:
$$g(d) = \left(1 - \frac{d}{R_E}\right) g_E$$.
Detailed Explanation
When we talk about points located below the Earth's surface, the concept changes slightly. Although the mass outside the radius of the smaller sphere does not exert gravitational force on the point mass, the mass within the sphere does exert a force as if it was concentrated at the center. Hence, gravitational acceleration increases linearly as we go deeper into the Earth. The formula presented shows that g(d) falls to a lower value as d increases, illustrating that gravity is weakest at the core of the Earth.
Examples & Analogies
Think of how a diver feels at different depths in water. Just like you can feel the pressure decrease as you swim to the surface, the force of gravity felt by firemen in a deep pit can similarly be thought of as diminishingβthe deeper you go, the less gravitational pull you experience from the Earth above.
General Observation
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Thus, the acceleration due to Earthβs gravity is maximum at its surface, decreasing whether you go up or down.
Detailed Explanation
This section summarizes the key takeaway: acceleration due to gravity is strongest at the Earth's surface, and it decreases both when ascending above the surface and when descending below the surface. The relationship described in this section introduces a unique characteristic of gravitational acceleration with elevation and depth.
Examples & Analogies
Consider a trampoline: when you are at the lowest point, you feel a force pushing you upwards; as you rise, this force lessens. Gravity acts similarly. Think of climbing to the edge of the Earth, reaching lower depths, or even in a rocketβyou will always feel gravity lessen as you travel away from or deeper into the Earth.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Acceleration due to Gravity: The force of gravity acting on an object based on its position.
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Reduction of Gravity at Height: Gravity decreases with increasing height above Earth's surface.
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Reduction of Gravity at Depth: Gravity decreases with increasing depth below Earth's surface.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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When an astronaut is in space, they feel less gravitational pull due to the increased height from Earth's surface.
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As a person descends into a mine, the gravitational pull they experience diminishes.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Up in the sky, gravity's shy, down below, it helps us flow.
π Fascinating Stories
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Imagine climbing a tall mountain, feeling lighter; when you dig into the Earth, it feels heavier at first, but then it lightens as you go deeper.
π§ Other Memory Gems
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H.E.L.P: Height Equals Lesser Pull.
π― Super Acronyms
G.R.A.D.E
- Gravity Reduces Above and Down Everywhere.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Height (h)
Definition:
The distance above the Earth's surface.
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Term: Depth (d)
Definition:
The distance below the Earth's surface.
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Term: Gravitational constant (G)
Definition:
A fundamental constant used in the calculation of gravitational force.
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Term: Mass (m)
Definition:
A measure of the amount of matter in an object.
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Term: Surface gravity (g)
Definition:
The acceleration due to gravity at the surface of the Earth.