The force, [tex]$F$[/tex], between two objects is inversely proportional to the square of the distance between them. The distance between two objects is increased by [tex]$50 \%$[/tex]. Calculate the percent reduction in the force between the objects.
Answer (2)
The force is inversely proportional to the square of the distance: F = d 2 k .
The distance increases by 50%: d n e w = 1.5 d .
The new force is F n e w = ( 1.5 d ) 2 k = 2.25 d 2 k .
The percent reduction in force is 9 5 × 100 ≈ 55.56% .
Explanation
Understanding Inverse Proportionality The force, F , between two objects is inversely proportional to the square of the distance between them. This means that F = d 2 k , where k is a constant and d is the distance between the objects.
Calculating New Distance The distance between the two objects is increased by 50% . If the original distance is d , the new distance is d n e w = d + 0.5 d = 1.5 d .
Calculating New Force The new force, F n e w , is given by F n e w = ( 1.5 d ) 2 k = 2.25 d 2 k .
Finding the Ratio of New Force to Old Force To find the percent reduction in the force, we need to calculate the ratio of the new force to the old force: F o l d F n e w = d 2 k 2.25 d 2 k = 2.25 1 = 4 9 1 = 9 4 .
Calculating Fractional Reduction The reduction in force is F o l d − F n e w . The fractional reduction in force is F o l d F o l d − F n e w = 1 − F o l d F n e w = 1 − 9 4 = 9 5 .
Calculating Percentage Reduction To find the percentage reduction, we multiply the fractional reduction by 100: 9 5 × 100 = 55.555...% . Rounding to two decimal places, the percent reduction in the force is approximately 55.56% .
Final Answer Therefore, the percent reduction in the force between the objects is approximately 55.56% .
Examples
Imagine you are adjusting the distance between a satellite and Earth. The gravitational force between them is inversely proportional to the square of the distance. If you increase the satellite's distance from Earth by 50%, you can calculate the reduction in the gravitational force using this concept. This helps in planning orbital adjustments and predicting changes in communication signal strength.
Increasing the distance between two objects by 50% causes the force between them to decrease by approximately 55.56%. This calculation is based on the inverse square law of force. By applying the appropriate mathematical steps, we can clearly see how distance affects force in this context.
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