Question

force of gravitational attraction between 2 objects = (universal gravitation constant) ((mass of 1st object mass of 2nd object) / (distance between centers of objects)²) ( f_g = g left( \frac{m_1 m_2}{d^2}
ight) ) where ( g = ) universal gravitation constant ( = 6.67 \times 10^{-11} \frac{n cdot m^2}{kg^2} ) the proportionalities expressed by newton’s universal law of gravitation are represented graphically by the following illustration. observe how the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. ( f_g = \frac{g m_1 m_2}{d^2} ) (graph of force vs distance, curve), ( f_g = \frac{g m_1 m_2}{d^2} ) (graph of force vs mass, line) effect of mass on ( f_{grav} ) (diagrams with masses and forces), effect of distance on ( f_{grav} ) (diagrams with distances and forces) 10) if the separation distance between the object and the earth is... a. increased by a factor of 2, the ( f_g ) is ______ by a factor of ______. b. increased by a factor of 3, the ( f_g ) is ______ by a factor of ______. c. reduced by half, the ( f_g ) is ______ by a factor of ______ 11) if the mass of the object is... d. increased by a factor of 2, the ( f_g ) is ______ by a factor of ______ e. increased by a factor of 3, the ( f_g ) is ______ by a factor of ______ f. reduced by half, the ( f_g ) is ______ by a factor of ______

Explanation:

Step1: Recall the formula for gravitational force

The formula for the gravitational force between two objects is \( F_g = G\frac{m_1m_2}{d^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( d \) is the distance between their centers.

Step2: Analyze the effect of distance on \( F_g \) (part a)

• Original force: \( F_{g1} = G\frac{m_1m_2}{d_1^2} \)
• New distance: \( d_2 = 2d_1 \)
• New force: \( F_{g2} = G\frac{m_1m_2}{d_2^2} = G\frac{m_1m_2}{(2d_1)^2} = G\frac{m_1m_2}{4d_1^2} = \frac{1}{4} \times G\frac{m_1m_2}{d_1^2} = \frac{1}{4}F_{g1} \)
• So, when the distance is increased by a factor of 2, the force is decreased by a factor of 4.

Step3: Analyze the effect of distance on \( F_g \) (part b)

• Original force: \( F_{g1} = G\frac{m_1m_2}{d_1^2} \)
• New distance: \( d_2 = 3d_1 \)
• New force: \( F_{g2} = G\frac{m_1m_2}{d_2^2} = G\frac{m_1m_2}{(3d_1)^2} = G\frac{m_1m_2}{9d_1^2} = \frac{1}{9} \times G\frac{m_1m_2}{d_1^2} = \frac{1}{9}F_{g1} \)
• So, when the distance is increased by a factor of 3, the force is decreased by a factor of 9.

Step4: Analyze the effect of distance on \( F_g \) (part c)

• Original force: \( F_{g1} = G\frac{m_1m_2}{d_1^2} \)
• New distance: \( d_2 = \frac{1}{2}d_1 \)
• New force: \( F_{g2} = G\frac{m_1m_2}{d_2^2} = G\frac{m_1m_2}{(\frac{1}{2}d_1)^2} = G\frac{m_1m_2}{\frac{1}{4}d_1^2} = 4 \times G\frac{m_1m_2}{d_1^2} = 4F_{g1} \)
• So, when the distance is reduced by half, the force is increased by a factor of 4.

Step5: Analyze the effect of mass on \( F_g \) (part d)

• Assume the mass of the object (say \( m_1 \)) is increased by a factor of 2, so \( m_{1new} = 2m_1 \), and \( m_2 \) (mass of Earth) remains constant.
• Original force: \( F_{g1} = G\frac{m_1m_2}{d^2} \)
• New force: \( F_{g2} = G\frac{(2m_1)m_2}{d^2} = 2 \times G\frac{m_1m_2}{d^2} = 2F_{g1} \)
• So, when the mass of the object is increased by a factor of 2, the force is increased by a factor of 2.

Step6: Analyze the effect of mass on \( F_g \) (part e)

• Assume the mass of the object (say \( m_1 \)) is increased by a factor of 3, so \( m_{1new} = 3m_1 \), and \( m_2 \) (mass of Earth) remains constant.
• Original force: \( F_{g1} = G\frac{m_1m_2}{d^2} \)
• New force: \( F_{g2} = G\frac{(3m_1)m_2}{d^2} = 3 \times G\frac{m_1m_2}{d^2} = 3F_{g1} \)
• So, when the mass of the object is increased by a factor of 3, the force is increased by a factor of 3.

Step7: Analyze the effect of mass on \( F_g \) (part f)

• Assume the mass of the object (say \( m_1 \)) is reduced by half, so \( m_{1new} = \frac{1}{2}m_1 \), and \( m_2 \) (mass of Earth) remains constant.
• Original force: \( F_{g1} = G\frac{m_1m_2}{d^2} \)
• New force: \( F_{g2} = G\frac{(\frac{1}{2}m_1)m_2}{d^2} = \frac{1}{2} \times G\frac{m_1m_2}{d^2} = \frac{1}{2}F_{g1} \)
• So, when the mass of the object is reduced by half, the force is decreased by a factor of \( \frac{1}{2} \) (or increased by a factor of \( \frac{1}{2} \), but since it's a reduction, we say decreased by a factor of 2).

Answer:

10)

a. decreased, \( \frac{1}{4} \)
b. decreased, \( \frac{1}{9} \)
c. increased, \( 4 \)

11)

d. increased, \( 2 \)
e. increased, \( 3 \)
f. decreased, \( \frac{1}{2} \)