Two forces who's resultant is 10N,are perpendicular to each other if one of them make an angle 60 degree with their resultant. Calculate its magnitude.
HOW
Let's call the magnitude of the first force F₁ and the magnitude of the second force F₂.
We are given that the resultant of the two forces is 10 N.
Since the two forces are perpendicular to each other, we can use the Pythagorean theorem to relate the magnitudes of the two forces and the magnitude of the resultant:
Resultant² = F₁² + F₂²
Given that the resultant is 10 N, we have:
10² = F₁² + F₂²
Simplifying this equation:
100 = F₁² + F₂² ----(1)
We are also given that one of the forces makes an angle of 60 degrees with their resultant. Let's call this force F₁.
To determine the magnitude of this force, we can use trigonometry. The component of force F₁ that is perpendicular to the resultant can be found using the equation:
F₁⊥ = F₁ * sin(60°) = F₁ * √3/2
Now we can substitute this value for F₁⊥ and the value of the resultant into equation (1):
100 = (F₁ * √3/2)² + F₂²
Simplifying this equation:
100 = 3/4 * F₁² + F₂²
Now, we have two equations with two unknowns. We can solve these equations simultaneously to find the magnitudes of F₁ and F₂.
From equation (1):
100 = F₁² + F₂²
From the previous equation:
100 = 3/4 * F₁² + F₂²
Multiply both sides of the second equation by 4/3:
(4/3) * 100 = F₁² + (4/3) * F₂²
Simplifying further:
(400/3) = F₁² + (4/3) * F₂²
Subtracting the first equation from this equation, we have:
(400/3) - 100 = F₁² + (4/3) * F₂² - (F₁² + F₂²)
Simplifying:
(400/3) - 100 = -F₂² + (4/3) * F₂²
(400/3) - 100 = (4/3 - 1) * F₂²
(400/3) - 100 = (1/3) * F₂²
(400/3) - 100 = (1/3) * F₂²
Multiplying both sides by 3:
400 - 300 = F₂²
100 = F₂²
Taking the square root of both sides:
F₂ = √100
F₂ = 10 N
So the magnitude of the second force, F₂, is 10 N.
Substituting this value of F₂ into equation (1):
100 = F₁² + 10²
100 = F₁² + 100
Subtracting 100 from both sides:
0 = F₁²
F₁ = 0 N
So the magnitude of the first force, F₁, is 0 N.
Therefore, the magnitude of the force that makes an angle of 60 degrees with the resultant is 0 N.
To solve this problem, we can use vector addition and basic trigonometry. Let's break it down step by step:
1. Let's assume the two forces are F1 and F2, and their resultant (or sum) is 10N. We need to find the magnitude of one of the forces.
2. Since the forces are perpendicular to each other, we can use the Pythagorean theorem to find the magnitude of F1 or F2. According to the theorem, the square of the hypotenuse (the resultant) is equal to the sum of the squares of the other two sides (forces):
(resultant)^2 = (F1)^2 + (F2)^2
(10N)^2 = (F1)^2 + (F2)^2
Simplifying:
100N^2 = F1^2 + F2^2
3. Next, we are given that one of the forces (let's assume it is F1) makes an angle of 60 degrees with the resultant. We can use trigonometry to find the relationship between F1, F2, and the angle.
In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse.
sin(angle) = (side opposite angle) / (hypotenuse)
sin(60°) = F1 / 10N
Rearranging the equation:
F1 = 10N * sin(60°)
Using a calculator:
F1 = 10N * 0.866 (approximately)
F1 ≈ 8.66N
So, the magnitude of one of the forces is approximately 8.66N.
Tan 60=opp/adj
tan 60= 10/10
Tan =1/1.7