Question

example #19: universal gravitation
a physics student is flying on an airplane 11,887 m above ground.
(a) determine the force of gravitational attraction between the earth (m = 5.98 x 10^24 kg) and the 70 kg physics student if the distance of earth’s radius is 6.38 x 10^6 m.
f_g = \frac{gm_ym_m}{r^2}
f_g = \frac{(6.67 \times 10^{-11})(70)(5.98 \times 10^{24})}{((6.38 \times 10^6) + 11,887)^2}
= 683.4 n
(b) find the acceleration of gravity that the student experiences.
f_g = m_a g
683.4 = (70)a_g
a_g = \frac{683.4}{70}
= 9.76 m/s^2
(c) if the student had more mass, how would this affect the acceleration the students experiences on the plane?

Explanation:

Step1: Calculate the distance from the center of the Earth

The distance \( r \) from the center of the Earth to the student is the sum of the Earth's radius \( R = 6.38\times10^{6}\text{ m} \) and the height \( h=11887\text{ m} \).
\( r=(6.38\times 10^{6}+11887)\text{ m}\approx6.392\times 10^{6}\text{ m} \)

Step2: Use the law of universal gravitation

The formula for the gravitational force is \( F_{g}=G\frac{Mm}{r^{2}} \), where \( G = 6.67\times10^{-11}\text{ N}\cdot\text{m}^{2}/\text{kg}^{2} \), \( M = 5.98\times10^{24}\text{ kg} \), and \( m = 70\text{ kg} \).

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Step3: Calculate the acceleration due to gravity

Using \( F_{g}=ma_{g} \), we can solve for \( a_{g} \). Given \( F_{g} = 683.4\text{ N} \) and \( m = 70\text{ kg} \), then \( a_{g}=\frac{F_{g}}{m} \)

$$ a_{g}=\frac{683.4}{70}=9.76\text{ m/s}^{2} $$

Step4: Analyze the effect of mass on acceleration

From \( F_{g}=G\frac{Mm}{r^{2}} \) and \( F_{g}=ma_{g} \), we can equate \( G\frac{Mm}{r^{2}}=ma_{g} \). The mass \( m \) of the student cancels out (\( a_{g}=G\frac{M}{r^{2}} \)). So if the student had more mass, the acceleration \( a_{g} \) that the student experiences would remain the same.

Answer:

(A) The force of gravitational attraction is \( 683.4\text{ N} \).
(B) The acceleration of gravity is \( 9.76\text{ m/s}^{2} \).
(C) The acceleration the student experiences would remain the same.