Calculate the change in the value of g at altitude 45 degree. Take radius of earth = 6370 km.

To calculate the change in the value of gravitational acceleration (g) at an altitude of 45 degrees latitude, we need to consider both the altitude and the effects of the Earth's rotation. The gravitational acceleration at the surface of the Earth is approximately 9.81 m/s², but this value decreases with altitude and varies slightly with latitude due to the Earth's shape and rotation.

Understanding Gravitational Acceleration

The gravitational force experienced by an object is influenced by two main factors: the distance from the center of the Earth and the latitude. The formula for gravitational acceleration (g) at a distance r from the center of the Earth is given by:

g = G * (M / r²)

Where:

• G is the gravitational constant (approximately 6.674 × 10⁻¹¹ N(m/kg)²),
• M is the mass of the Earth (about 5.972 × 10²⁴ kg),
• r is the distance from the center of the Earth.

Calculating the Radius at Altitude

At sea level, the radius of the Earth is approximately 6370 km. To find the radius at an altitude of 45 degrees latitude, we need to account for the altitude above sea level. However, since the question does not specify an altitude in kilometers, we will assume we are calculating g at sea level first and then consider the altitude effect.

Effect of Latitude on Gravitational Acceleration

The value of g also varies with latitude due to the Earth's rotation and its equatorial bulge. At the equator, g is slightly less than at the poles. The formula to adjust for latitude is:

g(latitude) = g₀ * (1 - 0.002637 * cos(2 * latitude))

Where g₀ is the standard gravitational acceleration (9.81 m/s²). For 45 degrees latitude, we can calculate:

Calculating g at 45 Degrees Latitude

Substituting the values:

g(45°) = 9.81 * (1 - 0.002637 * cos(90°))

Since cos(90°) = 0, the equation simplifies to:

g(45°) = 9.81 m/s²

Considering Altitude

If we were to consider an altitude above sea level, we would need to adjust the radius. For example, if we were at an altitude of 10 km, the new radius would be:

r = 6370 km + 10 km = 6380 km = 6,380,000 m

Then we would recalculate g using the formula:

g = G * (M / r²)

Final Calculation

Using the values:

g = 6.674 × 10⁻¹¹ N(m/kg)² * (5.972 × 10²⁴ kg / (6,380,000 m)²)

This gives us a value slightly less than 9.81 m/s², which decreases with altitude.

Summary

At sea level and at 45 degrees latitude, the value of g is approximately 9.81 m/s². If you were to ascend to a significant altitude, you would see a decrease in this value. The exact change would depend on the specific altitude you are considering. For practical purposes, at 10 km altitude, g would be around 9.77 m/s², illustrating how gravitational acceleration diminishes with height.