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ScanSolve AI Answer
To solve this problem, we need to find the point between the Earth and the Moon where the gravitational fields created by each body cancel each other out. Step 1: Define variables and the condition for a zero resultant gravitational field. Let M_E be the mass of the Earth and M_M be the mass of the Moon. We are given M_E = 80 M_M. Let D be the distance between the centers of the Earth and the Moon, D = 4.0 × 10^5 km. Let x be the distance from the center of the Earth to the point where the resultant gravitational field is zero. The distance from the center of the Moon to this point will then be D - x. The gravitational field g due to a mass M at a distance r is given by the formula g = (GM)/(r^2), where G is the gravitational constant. For the resultant field to be zero, the magnitudes of the gravitational fields from the Earth (g_E) and the Moon (g_M) must be equal: g_E = g_M. Step 2: Set up the equation for equal gravitational fields. (GM_E)/(x^2) = (GM_M)/((D-x)^2) Step 3: Substitute the masses and simplify the equation. We can cancel the gravitational constant G from both sides. (M_E)/(x^2) = (M_M)/((D-x)^2) Using the relationship M_E = 80 M_M: (80 M_M)/(x^2) = (M_M)/((D-x)^2) We can cancel M_M from both sides: (80)/(x^2) = (1)/((D-x)^2) Step 4: Solve the equation for the distance x. Take the square root of both sides (we consider the positive root since x is a distance): sqrt((80)/(x^2)) = sqrt((1)/((D-x)^2)) sqrt(80)x = (1)/(D-x) We know that sqrt(80) = sqrt(16 × 5) = 4sqrt(5). 4sqrt(5)x = (1)/(D-x) Cross-multiply to isolate x: 4sqrt(5)(D-x) = x 4sqrt(5)D - 4sqrt(5)x = x Group the terms with x: 4sqrt(5)D = x + 4sqrt(5)x 4sqrt(5)D = x(1 + 4sqrt(5)) x = 4sqrt(5)D1 + 4sqrt(5) Step 5: Calculate the numerical value. Substitute the value of D = 4.0 × 10^5 km: x = 4sqrt(5)1 + 4sqrt(5) × (4.0 × 10^5 km) Calculate the numerical value of 4sqrt(5)1 + 4sqrt(5): 4sqrt(5) ≈ 4 × 2.236067977 ≈ 8.944271908 1 + 4sqrt(5) ≈ 1 + 8.944271908 ≈ 9.944271908 (8.944271908)/(9.944271908) ≈ 0.900040008 Now, multiply by D: x ≈ 0.900040008 × 4.0 × 10^5 km x ≈ 3.600160032 × 10^5 km Rounding to two significant figures, as per the given D = 4.0 × 10^5 km: x ≈ 3.6 × 10^5 km The distance from the center of the Earth where the resultant gravitational field is zero is: 3.6 × 10^5 km That's 2 down. 3 left today — send the next one.