Two forces of magnitude 10 Newton and 12 newton on a particle if the angle between them is 60°,find the magnitude and direction of their resultant
To find the magnitude and direction of the resultant of two forces, you can use the law of cosines.
Step 1: Define the given forces and angle:
Force1 = 10 N
Force2 = 12 N
Angle = 60°
Step 2: Calculate the magnitude of the resultant force using the law of cosines:
Resultant^2 = Force1^2 + Force2^2 - 2 * Force1 * Force2 * cos(Angle)
Resultant^2 = 10^2 + 12^2 - 2 * 10 * 12 * cos(60°)
Resultant^2 = 100 + 144 - 240 * 0.5
Resultant^2 = 100 + 144 - 120
Resultant^2 = 124
Taking the square root of both sides:
Resultant = √124
Resultant ≈ 11.135 N
Step 3: Calculate the direction of the resultant force using the law of sines:
sin(θ) / Force1 = sin(180° - θ) / Resultant
sin(θ) / 10 = sin(180° - 60°) / 11.135
sin(θ) / 10 = sin(120°) / 11.135
sin(θ) / 10 = √3 / 11.135
Cross-multiply:
sin(θ) = (√3 / 11.135) * 10
sin(θ) = √3 / 1.1135
Taking the inverse sine:
θ = sin^(-1)(√3 / 1.1135)
θ ≈ 63.43°
Therefore, the magnitude of the resultant force is approximately 11.135 N, and the direction is approximately 63.43°.
To find the magnitude of the resultant force, we can use the law of cosines. The law of cosines states that for a triangle with sides a, b, and c and an angle θ across from side c:
c^2 = a^2 + b^2 - 2ab*cos(θ)
In this case, the magnitudes of the forces are a = 10 N and b = 12 N, and the angle between them is 60°. Let's call the magnitude of the resultant force c.
c^2 = 10^2 + 12^2 - 2*10*12*cos(60°)
c^2 = 100 + 144 - 240*cos(60°)
c^2 = 244 - 240(0.5)
c^2 = 244 - 120
c^2 = 124
c = square root of 124
c ≈ 11.14 N
Therefore, the magnitude of the resultant force is approximately 11.14 N.
To find the direction of the resultant force, we can use the law of sines. The law of sines states that for a triangle with sides a, b, and c and opposite angles A, B, and C:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, the magnitudes of the forces are a = 10 N and b = 12 N, and the angle opposite to the side c is 60°. Let's call the angle opposite to the side c θ.
10/sin(θ) = 12/sin(60°)
10*sin(60°) = 12*sin(θ)
10*(√3/2) = 12*sin(θ)
5√3 = 12*sin(θ)
sin(θ) = (5√3)/12
θ = arcsin((5√3)/12)
θ ≈ 34.43°
Therefore, the direction of the resultant force is approximately 34.43°.