Forces of 675 lb and 828 ob act on a body. The smaller force acts due north; the larger force acts N 52.3 degrees E. Find the direction and the magnitude of the resultant.
Set 'em up and knock 'em down:
use the law of cosines to find the resultant r:
r^2 = 675^2 + 828^2 - 2(675)(828)cos(180-52.3)
Having r now, you can use the law of sines to get the resultant angle a:
r/sin(180-52.3) = 828/sin a
where a is measured in degrees east of north.
To find the direction and magnitude of the resultant force, we can use vector addition.
First, let's represent the two forces as vectors. The smaller force of 675 lb acting due north can be represented as a vector with a magnitude of 675 lb and a direction of 0 degrees (since it acts due north).
The larger force of 828 lb acting N 52.3 degrees E can be represented as a vector with a magnitude of 828 lb and a direction of 52.3 degrees measured from north.
Next, we can resolve the larger force into its vertical and horizontal components using trigonometry. The vertical component can be found by multiplying the magnitude of the force by the sine of the angle, and the horizontal component can be found by multiplying the magnitude by the cosine of the angle.
Vertical component = 828 lb * sin(52.3 degrees)
Horizontal component = 828 lb * cos(52.3 degrees)
Now, let's denote the vertical component as Fv and the horizontal component as Fh.
Fv = 828 lb * sin(52.3 degrees)
Fh = 828 lb * cos(52.3 degrees)
To find the resultant force, we need to add the horizontal component of the larger force to the smaller force, and the vertical component of the larger force to the smaller force.
Resultant force in the vertical direction = Fv (from the larger force) + 675 lb (from the smaller force)
Resultant force in the horizontal direction = Fh (from the larger force)
Now, we can find the magnitude and direction of the resultant force using the Pythagorean theorem and inverse trigonometric functions.
Magnitude of the resultant force = sqrt((Resultant force in the horizontal direction)^2 + (Resultant force in the vertical direction)^2)
Direction of the resultant force = arctan((Resultant force in the vertical direction) / (Resultant force in the horizontal direction))
Plug in the values into the equations and perform the calculations to find the magnitude and direction of the resultant force.