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Chapter 4: Problem 47

Find the altitudes above the Earth's surface where Earth's gravitational field strength would be (a) two thirds and (b) one third of its value at the surface. [Hint: First find the radius for each situation; then recall that the altitude is the distance from the surface to a point above the surface. Use proportional reasoning.]

Short Answer

Expert verified
Answer: To find the altitudes where the gravitational field strength is two-thirds and one-third of its value at the surface, use the formula for gravitational field strength involving the gravitational constant (G), the Earth's mass (M), and the radius (r). Calculate the radii for two-thirds (r_(2/3)) and one-third (r_(1/3)) of the Earth's surface gravitational field strength using proportional reasoning. Finally, subtract the Earth's radius to find the altitudes, resulting in Altitude_(2/3) = r_(2/3) - Earth's radius and Altitude_(1/3) = r_(1/3) - Earth's radius.

Step by step solution

01

Recall the formula for Earth's gravitational field strength

The gravitational field strength (g) at distance r from the center of the Earth is given by: g = GM/r^2, where G is the gravitational constant (approximately 6.674 x 10^-11 Nm^2/kg^2), M is the Earth's mass (approximately 5.972 x 10^24 kg), and r is the radius at a given point.
02

Calculate the Earth's gravitational field strength at the surface

First, we need to find the gravitational field strength at the Earth's surface. The Earth's radius is approximately 6.371 x 10^6 meters. Using the formula from Step 1, the gravitational field strength at the surface (g_surface) is: g_surface = G*M/(Earth's radius)^2
03

Determine the gravitational field strength for each situation

Let's call the unknown radii for two thirds and one third of the surface gravitational field strength as r_(2/3) and r_(1/3), respectively. The gravitational field strength at the altitudes required are: g_(2/3) = (2/3) * g_surface g_(1/3) = (1/3) * g_surface
04

Calculate the radii for each situation using proportional reasoning

Since the ratio is fixed, we can use the relationship between the gravitational field strength at the surface and at the given altitudes to find the radii: g_surface / r^2 = g_(2/3) / r_(2/3)^2 g_surface / r^2 = g_(1/3) / r_(1/3)^2 Rearrange these equations to isolate r_(2/3) and r_(1/3): r_(2/3) = sqrt(1.5 * Earth's radius * r) r_(1/3) = sqrt(0.5 * Earth's radius * r)
05

Calculate the altitudes above the Earth's surface

Finally, we can find the altitudes by subtracting the Earth's radius. The altitude for two thirds and one third of the surface gravitational field strength are: Altitude_(2/3) = r_(2/3) - Earth's radius Altitude_(1/3) = r_(1/3) - Earth's radius

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