Question

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

Answer 1

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°.