Three forces N, 14N and 16N acting on a particle keep it in equilibrium. Find the angle between the forces 10N and 16N.
To find the angle between the forces 10N and 16N, we can use the fact that the particle is in equilibrium. This means that the net force acting on the particle is zero.
We can break down the three forces into their horizontal and vertical components. Let's call the angle between the 14N force and the horizontal axis as θ1, and the angle between the 16N force and the horizontal axis as θ2.
The horizontal components of the forces are:
N cos(θ1)
14N cos(0)
16N cos(θ2)
Since the particle is in equilibrium, the net horizontal force must be zero. So we have:
N cos(θ1) + 14N cos(0) + 16N cos(θ2) = 0
Simplifying this equation, we get:
N cos(θ1) + 16N cos(θ2) = -14N
We can do the same thing for the vertical components of the forces:
N sin(θ1)
14N sin(0)
16N sin(θ2)
Since the particle is in equilibrium, the net vertical force must be zero. So we have:
N sin(θ1) + 14N sin(0) + 16N sin(θ2) = 0
Simplifying this equation, we get:
N sin(θ1) + 16N sin(θ2) = 0
We can now solve these two equations for N and sin(θ1):
N = -16N cos(θ2) / cos(θ1)
sin(θ1) = -16 sin(θ2) / cos(θ1)
Now, we can use the Pythagorean theorem to find the magnitude of the 10N force:
sqrt((10N)^2 + (N sin(θ1))^2) = sqrt((10N)^2 + (16 sin(θ2))^2)
Finally, we can use the arctangent function to find the angle between the 10N and 16N forces:
tan(θ10-16) = (N sin(θ1)) / (10N) = (-16 sin(θ2)) / (10N cos(θ2))
θ10-16 = arctan((-16 sin(θ2)) / (10N cos(θ2)))
Therefore, we can find the angle between the forces 10N and 16N by using the above formula with the values we have calculated.