F = 8 Gm₁m₂ 2 r Two masses, m₁ and m₂ are separated by a distance, r. The force of attraction between the two masses is F. A) If m₁ increases by 9, how does F change? B) If r is halved (1/2), how would F change? C) If r is not changed but both masses increase by a factor 7, how would F change? D) If all variables (r, m₁ and m₂) increase by 4, how would F change?
F = 8 Gm₁m₂ 2 r Two masses, m₁ and m₂ are separated by a distance, r. The force of attraction between the two masses is F. A) If m₁ increases by 9, how does F change? B) If r is halved (1/2), how would F change? C) If r is not changed but both masses increase by a factor 7, how would F change? D) If all variables (r, m₁ and m₂) increase by 4, how would F change?
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![### Understanding Gravitational Force
The gravitational force (\( F_g \)) between two masses is described by the equation:
\[ F_g = \frac{G m_1 m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance separating the two masses.
Let's analyze how changes in the variables affect the force of attraction \( F \):
#### A) If \( m_1 \) increases by 9, how does \( F \) change?
If \( m_1 \) is increased by a factor of 9, the new mass \( m_1' \) = 9 \( m_1 \). The gravitational force will change as follows:
\[ F_g' = \frac{G \cdot (9m_1) \cdot m_2}{r^2} = 9 \cdot \frac{G m_1 m_2}{r^2} \]
Hence, \( F \) increases by a factor of 9.
#### B) If \( r \) is halved (\(\frac{1}{2}\)), how would \( F \) change?
If the distance \( r \) is halved, the new distance \( r' \) = \( \frac{r}{2} \). The new gravitational force will be:
\[ F_g' = \frac{G m_1 m_2}{(\frac{r}{2})^2} = \frac{G m_1 m_2}{\frac{r^2}{4}} = 4 \cdot \frac{G m_1 m_2}{r^2} \]
Thus, \( F \) increases by a factor of 4.
#### C) If \( r \) is not changed but both masses increase by a factor of 7, how would \( F \) change?
If both \( m_1 \) and \( m_2 \) increase by a factor of 7, the new masses are \( m_1' \) = 7 \( m_1 \) and \( m_2' \) = 7 \( m_2 \). The new gravitational force will be:
\[ F_g' = \frac{G \cdot (7m_1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbb95dca4-f6c7-46fa-bbcb-5cdbe2ca0e09%2Fa6472281-05bd-4798-8ddc-6afa48162e76%2Fzi2wz3f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Gravitational Force
The gravitational force (\( F_g \)) between two masses is described by the equation:
\[ F_g = \frac{G m_1 m_2}{r^2} \]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses,
- \( r \) is the distance separating the two masses.
Let's analyze how changes in the variables affect the force of attraction \( F \):
#### A) If \( m_1 \) increases by 9, how does \( F \) change?
If \( m_1 \) is increased by a factor of 9, the new mass \( m_1' \) = 9 \( m_1 \). The gravitational force will change as follows:
\[ F_g' = \frac{G \cdot (9m_1) \cdot m_2}{r^2} = 9 \cdot \frac{G m_1 m_2}{r^2} \]
Hence, \( F \) increases by a factor of 9.
#### B) If \( r \) is halved (\(\frac{1}{2}\)), how would \( F \) change?
If the distance \( r \) is halved, the new distance \( r' \) = \( \frac{r}{2} \). The new gravitational force will be:
\[ F_g' = \frac{G m_1 m_2}{(\frac{r}{2})^2} = \frac{G m_1 m_2}{\frac{r^2}{4}} = 4 \cdot \frac{G m_1 m_2}{r^2} \]
Thus, \( F \) increases by a factor of 4.
#### C) If \( r \) is not changed but both masses increase by a factor of 7, how would \( F \) change?
If both \( m_1 \) and \( m_2 \) increase by a factor of 7, the new masses are \( m_1' \) = 7 \( m_1 \) and \( m_2' \) = 7 \( m_2 \). The new gravitational force will be:
\[ F_g' = \frac{G \cdot (7m_1
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