Select the correct answer.
A force of 5.25 newtons acts on an object of a mass 25.5 kilograms. How far is the object from the center of Earth? (The value of G is [tex]$6.673 \times 10^{-11}$[/tex] newton meter [tex]$^2 /$[/tex] kilogram [tex]$^2$[/tex]. The mass of Earth is [tex]$5.98 \times 10^{24}$[/tex] kilograms.)
A. [tex]$2.1 \times 10^7$[/tex] meters
B. [tex]$4.4 \times 10^7$[/tex] meters
C. [tex]$5.8 \times 10^7$[/tex] meters
D. [tex]$6.9 \times 10^7$[/tex] meters
Answer (1)
The gravitational force formula is F = G r 2 M m .
Rearrange the formula to solve for r : r = F GM m .
Substitute the given values: r = 5.25 ( 6.673 × 1 0 − 11 ) ( 5.98 × 1 0 24 ) ( 25.5 ) .
Calculate r : r = 4.4 × 1 0 7 meters. The final answer is 4.4 × 1 0 7
Explanation
Understanding the Problem We are given the force acting on an object, its mass, the gravitational constant, and the mass of the Earth. We need to find the distance of the object from the center of the Earth.
Stating the Formula The gravitational force between two objects is given by the formula: F = G r 2 M m where:
F is the gravitational force,
G is the gravitational constant,
M is the mass of the Earth,
m is the mass of the object,
r is the distance between the centers of the two objects.
Rearranging the Formula We need to rearrange the formula to solve for r :
r = F GM m
Substituting the Values Now, we substitute the given values into the formula: r = 5.25 N ( 6.673 × 1 0 − 11 N m 2 / kg 2 ) ( 5.98 × 1 0 24 kg ) ( 25.5 kg )
Calculating the Distance Calculating the value of r :
r = 5.25 6.673 × 1 0 − 11 × 5.98 × 1 0 24 × 25.5 r = 5.25 1.016 × 1 0 15 r = 1.935 × 1 0 14 r = 4.4 × 1 0 7 meters
Selecting the Correct Answer Comparing the calculated value of r with the given options, we find that the correct answer is: B. 4.4 × 1 0 7 meters
Examples
Understanding gravitational forces is crucial in many real-world applications. For example, calculating the orbits of satellites, understanding tides, and even designing structures that can withstand the forces of gravity all rely on this principle. By understanding how gravitational force changes with distance, engineers can accurately predict the behavior of objects in space and on Earth.
