The resultant of two forces is 50N. If the
forces are perpendicular to each other
and one of them makes an angle of 300
with the resultant, find its magnitude
Gauaa
To find the magnitude of one of the forces, let's use the law of cosines.
In a triangle formed by the two forces and the resultant, we can label the forces as follows:
- Force 1: unknown magnitude, makes an angle of 30° with the resultant
- Force 2: unknown magnitude, makes an angle of 90° with the resultant
- Resultant force: 50N
Using the law of cosines, we have:
Resultant^2 = Force1^2 + Force2^2 - 2 * Force1 * Force2 * cosθ
where θ is the angle between Force1 and Force2. In this case, θ is 90°.
Plugging in the known values:
50^2 = Force1^2 + Force2^2 - 2 * Force1 * Force2 * cos(90°)
Simplifying:
2500 = Force1^2 + Force2^2 - 0
2500 = Force1^2 + Force2^2
Since Force2 makes an angle of 90° with the resultant, it can be considered as the hypotenuse of a right triangle.
By using the Pythagorean theorem, we can express Force2 in terms of its components:
Force2^2 = Force2x^2 + Force2y^2
Given that Force2 makes a right angle with the x and y axis, Force2x = 0 and Force2y = 50N.
Substituting these values:
2500 = Force1^2 + (Force2x^2 + Force2y^2)
2500 = Force1^2 + (0^2 + 50^2)
2500 = Force1^2 + 2500
Now, isolate Force1:
Force1^2 = 2500 - 2500
Force1^2 = 0
Taking the square root of both sides:
Force1 = √0
Therefore, the magnitude of Force1 is 0N.
To find the magnitude of one of the forces, given the resultant force and the angle it makes with one of the forces, we can use vector addition and trigonometry.
Let's assume that the magnitude of the first force is F1 and the magnitude of the second force is F2. Since the forces are perpendicular to each other, we can visualize them as the two legs of a right triangle. The resultant force (R) is the hypotenuse of this right triangle.
Using the Pythagorean theorem, we can express the relationship between the magnitudes of the forces and the resultant force:
R^2 = F1^2 + F2^2
Here, R = 50N (given in the question).
To find the magnitude of one of the forces (let's say F1), we need to find its value by rearranging the equation:
F1^2 = R^2 - F2^2
F1 = sqrt(R^2 - F2^2)
Now, we are given that one of the forces (F2) makes an angle of 30 degrees (300 converted to degrees) with the resultant. To find F2, we can use trigonometric functions (specifically, the sine function).
sin(θ) = Opposite / Hypotenuse
In this case, θ (angle) = 30 degrees and the hypotenuse (R) = 50N. The opposite side would be F2.
sin(30 degrees) = F2 / 50N
To solve for F2, we rearrange the equation:
F2 = sin(30 degrees) * 50N
Now that we have the value of F2, we can substitute it back into the equation to find F1:
F1 = sqrt(R^2 - F2^2)
F1 = sqrt(50N^2 - F2^2)
Finally, calculate the magnitude of F1 using the given values:
F1 = sqrt(50N^2 - (sin(30 degrees) * 50N)^2)
This will give you the magnitude of the force F1.
sketch it
put resultant along x axis
F1 is then at -60 degrees (we want F1)
F2 is at +30 degrees
so
F1 cos 60 + F2 cos 30 = 50
F1 sin 60 - F2 sin 30 = 0