two forces whose magnitude are in the ratio 3:5 give a resultant equal to 35 newton if the angle between them is 60 degrees find the magnitude if the resultant
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To find the magnitude of the resultant force, we can use vector addition.
Let's assume that the magnitude of the first force is 3x and the magnitude of the second force is 5x, where x is a scaling factor.
Now, we can use trigonometry to find the components of each force. The horizontal component of the first force can be found using the formula:
F₁x = F₁ * cosθ₁
Similarly, the vertical component of the first force can be found using:
F₁y = F₁ * sinθ₁
Where θ₁ is the angle between the first force and the x-axis.
Using the same logic, we can find the components of the second force:
F₂x = F₂ * cosθ₂
F₂y = F₂ * sinθ₂
Where θ₂ is the angle between the second force and the x-axis.
Since the angle between the two forces is 60 degrees, θ₁ and θ₂ are 60 degrees apart. This means that:
θ₂ = θ₁ + 60°
Let's substitute in the given values and solve for the components:
F₁ = 3x
F₂ = 5x
θ₁ = 60°
θ₂ = 60° + 60° = 120°
F₁x = 3x * cos60° = 3x * 0.5 = 1.5x
F₁y = 3x * sin60° = 3x * √3/2 = 1.5√3x
F₂x = 5x * cos120° = 5x * (-0.5) = -2.5x
F₂y = 5x * sin120° = 5x * √3/2 = 2.5√3x
Now, we can add the components to find the resultant components:
Resultant horizontal component, Rₓ = F₁x + F₂x = 1.5x - 2.5x = -1x
Resultant vertical component, Rᵧ = F₁y + F₂y = 1.5√3x + 2.5√3x = 4√3x
The magnitude of the resultant force, R, can be found using the Pythagorean theorem:
R = √(Rₓ² + Rᵧ²)
Substituting the values:
R = √((-1x)² + (4√3x)²)
R = √(x² + 48x²)
R = √(49x²)
R = 7x
We know that the magnitude of the resultant force is equal to 35 Newtons, so:
7x = 35
Solving for x:
x = 5
Now, substituting the value of x back into the equation for R:
R = 7x = 7 * 5 = 35
Therefore, the magnitude of the resultant force is 35 Newtons.