the magnitude of the resultant of two perpendicular forces is 50 newton .if the resultant makes with the first force an angle of measure 30°,find the magnitude of each of these two forces
sin 30 =Fy/50
cos 30 = Fx/50
To find the magnitude of each of the two forces, we can use trigonometric relationships.
Let's call the magnitude of the first force F1 and the magnitude of the second force F2.
Given that the resultant of the two perpendicular forces is 50 newtons, we can use the Pythagorean theorem:
(1) 50^2 = F1^2 + F2^2
We are also given that the resultant force makes an angle of 30° with the first force. This means we can use trigonometric functions to relate the angles and the magnitudes of the forces.
The angle between the resultant force and the second force is 90° - 30° = 60°.
Using trigonometric relationships, we can write:
(2) F1 = F2 × tan(30°)
(3) F2 = F1 × tan(60°)
Now we can solve for the magnitudes of the forces. Let's substitute equation (2) in equation (1):
50^2 = (F2 × tan(30°))^2 + F2^2
Simplifying:
2500 = F2^2 × tan(30°)^2 + F2^2
Let's factor out F2^2:
2500 = F2^2 × (tan(30°)^2 + 1)
tan(30°)^2 + 1 = (1/√3)^2 + 1 = 1/3 + 1 = 4/3
2500 = F2^2 × (4/3)
Now we can solve for F2:
F2^2 = 2500 × 3/4
F2^2 = 7500/4
F2^2 = 1875
Taking the square root of both sides:
F2 ≈ √1875 ≈ 43.30 newton
Now we can substitute this value back into equation (2) to find F1:
F1 = F2 × tan(30°)
F1 ≈ 43.30 × (√3/3)
F1 ≈ 25.00 newton
Therefore, the magnitude of the first force (F1) is approximately 25 newtons, and the magnitude of the second force (F2) is approximately 43.30 newtons.