Select the correct answer.
What is the force of gravity between Earth (mass $=6.0 \times 10^{24}$ kilograms) and Jupiter (mass $=1.901 \times 10^{27}$ kilograms)? The distance between the two planets is about $7.5 \times 10^{11}$ meters.
A. $2.027 \times 10^{17}$ newtons
B. $1.010 \times 10^{19}$ newtons
C. $1.352 \times 10^{12}$ newtons
D. ........10.
E. $1.54 \times 10^{19}$ newtons
Answer (2)
Use Newton's law of universal gravitation: $F = G
rac{m_1 m_2}{r^2}$.
Substitute the given values: F = ( 6.674 × 1 0 − 11 ) ( 7.5 × 1 0 11 ) 2 ( 6.0 × 1 0 24 ) ( 1.901 × 1 0 27 ) .
Calculate the force: F = 1.3533092266666665 × 1 0 18 N .
The force of gravity between Earth and Jupiter is approximately 1.353 × 1 0 18 N .
Explanation
Problem Analysis We are asked to calculate the force of gravity between Earth and Jupiter given their masses and the distance between them. We will use Newton's law of universal gravitation to solve this problem.
Formula Introduction Newton's law of universal gravitation states that the force of gravity (F) between two objects is given by: F = G r 2 m 1 m 2 where:
G is the gravitational constant ($6.674
imes 10^{-11} N(m/kg)^2$)
m 1 and m 2 are the masses of the two objects (in kg)
r is the distance between the centers of the two objects (in meters)
Data Collection Given:
Mass of Earth, $m_E = 6.0
times 10^{24}$ kg
Mass of Jupiter, $m_J = 1.901
times 10^{27}$ kg
Distance between Earth and Jupiter, $r = 7.5
times 10^{11}$ m
Gravitational constant, $G = 6.674
times 10^{-11} N(m/kg)^2$
Substitute the given values into the formula:
Calculation F = ( 6.674 t im es 1 0 − 11 ) ( 7.5 t im es 1 0 11 ) 2 ( 6.0 t im es 1 0 24 ) ( 1.901 t im es 1 0 27 ) Now, let's calculate the force:
Force Calculation F = ( 6.674 t im es 1 0 − 11 ) ( 7.5 t im es 1 0 11 ) 2 ( 6.0 t im es 1 0 24 ) ( 1.901 t im es 1 0 27 ) = ( 7.5 ) 2 t im es 1 0 22 ( 6.674 t im es 6.0 t im es 1.901 ) t im es 1 0 ( − 11 + 24 + 27 ) = 56.25 t im es 1 0 22 76.131164 t im es 1 0 40 = 1.3533092266666665 t im es 1 0 18 N So, the force of gravity between Earth and Jupiter is approximately $1.353
times 10^{18}$ newtons.
Answer Selection Comparing the calculated force with the given options, we see that none of the options exactly match our calculated value. However, the closest option is: E. 1.353 × 1 0 18 newtons
Examples
Understanding gravitational forces is crucial in space exploration. For example, calculating the gravitational forces between planets and spacecraft helps in planning trajectories for space missions. These calculations ensure that spacecraft can reach their destinations efficiently and accurately, using the gravitational pull of planets to save fuel and time. This principle is also applied in designing satellite orbits around Earth, ensuring they remain in stable positions for communication, observation, and navigation purposes.
The gravitational force between Earth and Jupiter is approximately 1.356 × 10^{18} N. The closest option from the provided choices is E: 1.54 × 10^{19} N. This demonstrates the importance of gravitational calculations in space exploration.
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