The acceleration due to gravity, g, is constant at sea level on the Earth's surface. However, the acceleration decreases as an object moves away from the Earth's surface due to the increase in distance from the center of the Earth. Derive an expression for the acceleration due to gravity at a distance h above the surface of the Earth, gh. Express the equation in terms of the radius R of the Earth, g, and h. &h= Suppose a 91.75 kg hiker has ascended to a height of 1802 m above sea level in the process of climbing Mt. Washington. By what percent has the hiker's weight changed from its value at sea level as a result of climbing to this elevation? Use g= 9.807 m/s² and R = 6.371 x 10 m. Enter your answer as a positive value. weight change a =
The acceleration due to gravity, g, is constant at sea level on the Earth's surface. However, the acceleration decreases as an object moves away from the Earth's surface due to the increase in distance from the center of the Earth. Derive an expression for the acceleration due to gravity at a distance h above the surface of the Earth, gh. Express the equation in terms of the radius R of the Earth, g, and h. &h= Suppose a 91.75 kg hiker has ascended to a height of 1802 m above sea level in the process of climbing Mt. Washington. By what percent has the hiker's weight changed from its value at sea level as a result of climbing to this elevation? Use g= 9.807 m/s² and R = 6.371 x 10 m. Enter your answer as a positive value. weight change a =
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![### Acceleration Due to Gravity at Different Heights
The acceleration due to gravity, \( g \), is constant at sea level on the Earth's surface. However, the acceleration decreases as an object moves away from the Earth's surface due to the increase in distance from the center of the Earth. Derive an expression for the acceleration due to gravity at a distance \( h \) above the surface of the Earth, \( g_h \). Express the equation in terms of the radius \( R \) of the Earth, \( g \), and \( h \).
\[ g_h = \]
---
### Weight Change at Different Elevations
Suppose a 91.75 kg hiker has ascended to a height of 1802 m above sea level in the process of climbing Mt. Washington. By what percent has the hiker's weight changed from its value at sea level as a result of climbing to this elevation? Use \( g = 9.807 \, \text{m/s}^2 \) and \( R = 6.371 \times 10^6 \, \text{m} \). Enter your answer as a positive value.
\[ \text{weight change} = \quad \boxed{} \, \% \]
### Detailed Explanation:
1. **Gravity at Height \( h \):**
To derive the expression for \( g_h \), start with Newton's law of gravitation:
\[
F = \frac{G M_1 M_2}{R^2}
\]
Where:
- \( F \) is the gravitational force
- \( G \) is the gravitational constant
- \( M_1 \) and \( M_2 \) are the masses
- \( R \) is the distance between the centers of two masses
For gravitational acceleration \( g \) at the Earth's surface:
\[
g = \frac{GM}{R^2}
\]
At height \( h \):
\[
g_h = \frac{GM}{(R + h)^2}
\]
2. **Weight Change Calculation:**
- Calculate the gravitational acceleration at the elevation of 1802 meters:
\[
g_{h} = g \left( \frac{R^2}{(R + h)^2} \right)
\]
- Determine the weight change percentage:
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Transcribed Image Text:### Acceleration Due to Gravity at Different Heights
The acceleration due to gravity, \( g \), is constant at sea level on the Earth's surface. However, the acceleration decreases as an object moves away from the Earth's surface due to the increase in distance from the center of the Earth. Derive an expression for the acceleration due to gravity at a distance \( h \) above the surface of the Earth, \( g_h \). Express the equation in terms of the radius \( R \) of the Earth, \( g \), and \( h \).
\[ g_h = \]
---
### Weight Change at Different Elevations
Suppose a 91.75 kg hiker has ascended to a height of 1802 m above sea level in the process of climbing Mt. Washington. By what percent has the hiker's weight changed from its value at sea level as a result of climbing to this elevation? Use \( g = 9.807 \, \text{m/s}^2 \) and \( R = 6.371 \times 10^6 \, \text{m} \). Enter your answer as a positive value.
\[ \text{weight change} = \quad \boxed{} \, \% \]
### Detailed Explanation:
1. **Gravity at Height \( h \):**
To derive the expression for \( g_h \), start with Newton's law of gravitation:
\[
F = \frac{G M_1 M_2}{R^2}
\]
Where:
- \( F \) is the gravitational force
- \( G \) is the gravitational constant
- \( M_1 \) and \( M_2 \) are the masses
- \( R \) is the distance between the centers of two masses
For gravitational acceleration \( g \) at the Earth's surface:
\[
g = \frac{GM}{R^2}
\]
At height \( h \):
\[
g_h = \frac{GM}{(R + h)^2}
\]
2. **Weight Change Calculation:**
- Calculate the gravitational acceleration at the elevation of 1802 meters:
\[
g_{h} = g \left( \frac{R^2}{(R + h)^2} \right)
\]
- Determine the weight change percentage:
\
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