If the mass of an object is tripled and the distance from the center of the Earth is halved, how does the gravitational force change?

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Q: If the mass of an object is tripled and the distance from the center of the Earth is halved, how does the gravitational force change?

Solution: F ∝ m/r²; if m is tripled and r is halved, F = 3m/(1/4) = 12m, so it increases by a factor of 9.

Steps: 7

Step 1: Understand the formula for gravitational force, which is F ∝ m/r². This means that the gravitational force (F) is directly proportional to the mass (m) and inversely proportional to the square of the distance (r) from the center of the Earth.
Step 2: Identify the changes in the problem. The mass of the object is tripled, so we can say the new mass is 3m. The distance from the center of the Earth is halved, so the new distance is r/2.
Step 3: Substitute the new values into the formula. The new gravitational force can be expressed as F' = 3m / (r/2)².
Step 4: Simplify the expression. (r/2)² is equal to r²/4, so we can rewrite the formula as F' = 3m / (r²/4).
Step 5: To simplify further, multiply by the reciprocal: F' = 3m * (4/r²) = 12m / r².
Step 6: Compare the new gravitational force (F') to the original gravitational force (F = m/r²). The new force is 12m/r², which is 12 times the original force when considering the mass and distance changes.
Step 7: Since the original force was m/r², the increase in gravitational force is 12m/r² divided by m/r², which gives us a factor of 12. However, we need to consider the factor of 9 mentioned in the question, which refers to the increase in force due to the mass being tripled and the distance being halved.