Answer :
To determine which of the given pairs of forces may be the rectangular components of a force of 10 Newtons, we will calculate the magnitude of the resultant force for each given pair using the Pythagorean theorem. Specifically, for components [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the resultant force [tex]\( R \)[/tex] is given by:
[tex]\[ R = \sqrt{x^2 + y^2} \][/tex]
The goal is to find the pair for which this resultant force [tex]\( R \)[/tex] is 10 Newtons.
Let's examine each option:
Option A: 6 Newtons and 8 Newtons
[tex]\[ R = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \][/tex]
Since the resultant force is 10 Newtons, option A is a valid pair.
Option B: 2.5 Newtons and 25 Newtons
[tex]\[ R = \sqrt{2.5^2 + 25^2} = \sqrt{6.25 + 625} = \sqrt{631.25} \][/tex]
The resultant force [tex]\( \sqrt{631.25} \)[/tex] is not equal to 10 Newtons. So, option B is not valid.
Option C: 2 Newtons and 5 Newtons
[tex]\[ R = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \][/tex]
The resultant force [tex]\( \sqrt{29} \)[/tex] is not equal to 10 Newtons. Hence, option C is also not valid.
Option D: 3 Newtons and 5 Newtons
[tex]\[ R = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
The resultant force [tex]\( \sqrt{34} \)[/tex] is not equal to 10 Newtons. Therefore, option D is not valid.
Based on the calculations, the rectangular components that result in a force of 10 Newtons are those given in option A: 6 Newtons and 8 Newtons.
[tex]\[ R = \sqrt{x^2 + y^2} \][/tex]
The goal is to find the pair for which this resultant force [tex]\( R \)[/tex] is 10 Newtons.
Let's examine each option:
Option A: 6 Newtons and 8 Newtons
[tex]\[ R = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \][/tex]
Since the resultant force is 10 Newtons, option A is a valid pair.
Option B: 2.5 Newtons and 25 Newtons
[tex]\[ R = \sqrt{2.5^2 + 25^2} = \sqrt{6.25 + 625} = \sqrt{631.25} \][/tex]
The resultant force [tex]\( \sqrt{631.25} \)[/tex] is not equal to 10 Newtons. So, option B is not valid.
Option C: 2 Newtons and 5 Newtons
[tex]\[ R = \sqrt{2^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \][/tex]
The resultant force [tex]\( \sqrt{29} \)[/tex] is not equal to 10 Newtons. Hence, option C is also not valid.
Option D: 3 Newtons and 5 Newtons
[tex]\[ R = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
The resultant force [tex]\( \sqrt{34} \)[/tex] is not equal to 10 Newtons. Therefore, option D is not valid.
Based on the calculations, the rectangular components that result in a force of 10 Newtons are those given in option A: 6 Newtons and 8 Newtons.
