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state-variation-with-altitude

🚀 Gravitation is a fundamental force of nature that attracts two bodies towards each other. The acceleration due to gravity is the rate at which an object accelerates towards the Earth due to this gravitational force. It is denoted by 'g' and has a standard value of approximately 9.81 m/s² at sea level. However, this value varies with altitude, as the gravitational force decreases with an increase in distance from the center of the Earth. This variation can be understood through the formula for gravitational acceleration, which is inversely proportional to the square of the distance from the center of the Earth.

Theory Explanation

Understanding Gravitational Force

The gravitational force between two masses is given by Newton's law of universal gravitation, which states that every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

\[ F = G \frac{m_1 m_2}{r^2} \]
Acceleration Due to Gravity

The acceleration due to gravity (g) at a distance r from the center of the Earth can be derived from the gravitational force formula. If we consider the mass of the Earth (M) and the radius of the Earth (R), the acceleration due to gravity at the surface is given by g = GM/R². As we move to a height h above the Earth's surface, the distance from the center becomes (R + h), and the formula becomes g' = GM/(R + h)².

\[ g' = g \frac{R^2}{(R + h)^2} \]
Variation of g with Altitude

As altitude increases, the value of g decreases. This is because the distance from the center of the Earth increases, leading to a decrease in gravitational force. The relationship can be expressed as g' = g (1 - 2h/R + ...), where h is the height above the Earth's surface and R is the radius of the Earth.

\[ g' = g \left(1 - \frac{2h}{R}\right) \]

Key Points

  • 🎯 Gravitational force is universal and acts between any two masses.
  • 🎯 The acceleration due to gravity decreases with an increase in altitude.
  • 🎯 At sea level, the average value of g is approximately 9.81 m/s².
  • 🎯 The variation of g with altitude can be calculated using the formula g' = g (1 - 2h/R).
  • 🎯 Understanding the concept of gravitational potential energy is essential for grasping the implications of g's variation.

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Examples:💡

Calculate the acceleration due to gravity at a height of 1000 m above the Earth's surface.

Solution:

Step 1: Given: Height (h) = 1000 m, Radius of Earth (R) = 6.4 x 10^6 m, g = 9.81 m/s². We use the formula g' = g (1 - 2h/R).

\[ g' = 9.81 \left(1 - \frac{2 \times 1000}{6.4 \times 10^6}\right) \]

Step 2: Calculate the value: g' = 9.81 \left(1 - \frac{2000}{6.4 \times 10^6}\right) = 9.81 \left(1 - 0.0003125\right) = 9.81 \times 0.9996875.

\[ g' \approx 9.81 \times 0.9996875 = 9.807 m/s². \]

Step 3: Thus, the acceleration due to gravity at 1000 m altitude is approximately 9.807 m/s².

Common Mistakes

  • Mistake: Students often forget to convert units when calculating height or radius, leading to incorrect results.

    Correction: Always ensure that all measurements are in consistent units, typically meters for height and radius.

  • Mistake: Some students assume that g remains constant at all altitudes, which is incorrect.

    Correction: Remember that g decreases with altitude; use the appropriate formula to calculate the variation.

  • Mistake: Misapplying the formula for g' by not considering the correct values for R and h.

    Correction: Double-check the values used in the formula and ensure they correspond to the correct physical quantities.

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