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Select the correct answer.

Object A attracts object B with a gravitational force of 5 newtons from a given distance. If the distance between the two objects is reduced by half, what will be the new force of attraction between them?

A. 2.5 newtons
B. 10 newtons
C. 15 newtons
D. 20 newtons
E. 25 newtons



Answer :

Let's solve this problem step by step.

1. Initial Information:
- Initial gravitational force (F_initial) between object A and object B is 5 newtons.
- The distance between the two objects is reduced to half of its original distance.

2. Law of Gravitation:
The gravitational force between two objects is inversely proportional to the square of the distance between them. This means:

[tex]\[ F \propto \frac{1}{d^2} \][/tex]

Where [tex]\( F \)[/tex] is the gravitational force and [tex]\( d \)[/tex] is the distance between the objects.

3. Initial Distance Concept:
Let's denote the initial distance as [tex]\( d \)[/tex]. The formula for the initial force can be written as:

[tex]\[ F_initial = \frac{G \times m_A \times m_B}{d^2} \][/tex]

Since the constants [tex]\( G \times m_A \times m_B \)[/tex] are not changing, we can focus on the distance part.

4. Changed Distance:
The new distance is half of the original distance, so the new distance [tex]\( d_{new} \)[/tex] is:

[tex]\[ d_{new} = \frac{d}{2} \][/tex]

5. Changed Force Calculation:
We substitute the new distance into the formula for the gravitational force:

[tex]\[ F_{new} = \frac{G \times m_A \times m_B}{(d_{new})^2} \][/tex]

Since [tex]\( d_{new} = \frac{d}{2} \)[/tex]:

[tex]\[ F_{new} = \frac{G \times m_A \times m_B}{\left(\frac{d}{2}\right)^2} = \frac{G \times m_A \times m_B}{\frac{d^2}{4}} = G \times m_A \times m_B \times \frac{4}{d^2} \][/tex]

Now the new force is:

[tex]\[ F_{new} = 4 \times \frac{G \times m_A \times m_B}{d^2} = 4 \times F_{initial} \][/tex]

6. Substituting the Initial Force:
We know the initial force [tex]\( F_{initial} \)[/tex] is 5 newtons:

[tex]\[ F_{new} = 4 \times 5 = 20 \, \text{newtons} \][/tex]

So, the changed force of attraction when the distance is reduced to half is 20 newtons.

The correct answer is:
O D. 20 newtons