Two forces of magnitudes 60 Newtons and 80 Newtons make an angle of 30� with each other and are
applied to an object at the same point. Find the following:
1. the magnitude of the resultant force, and
2. to the nearest degree the angle it makes with the force of 60 Newtons.
Fr = 60N @ 30o,CCW + 80N @ 60o,CCW =
Resultant force.
60o - 30o = 30o Bet. the forces.
1. X = 60*cos30 + 80*cos60 = 92 N.
Y = 60*sin30 + 80*sin60 = 99.3 N.
Fr^2 = X^2 + Y^2 = 18,324,49
Fr = 135.4 N = Resultant force.
2. tanA = Y/X = 99.3/92 = 1.0130
A = 45.4o,CCW = Resultant angle.
45.3 - 30 = 15o Bet. Fr and the 60N force.
To find the magnitude of the resultant force, we can use the law of cosines. This law states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides multiplied by the cosine of the included angle.
In this case, we have a triangle with sides of 60 Newtons and 80 Newtons, and the included angle is 30 degrees.
Let's use the law of cosines to find the magnitude of the resultant force:
a² = b² + c² - 2bc * cos(A)
where:
a is the magnitude of the resultant force
b is the magnitude of the first force (60 N)
c is the magnitude of the second force (80 N)
A is the included angle (30 degrees)
Plugging in the values:
a² = (60)² + (80)² - 2(60)(80) * cos(30)
a² = 3600 + 6400 - 9600 * 0.866
a² = 3600 + 6400 - 8329.6
a² = 1670.4
Taking the square root of both sides, we get:
a ≈ 40.87
Therefore, the magnitude of the resultant force is approximately 40.87 Newtons.
Now, to find the angle that the resultant force makes with the force of 60 Newtons, we can use the law of sines. This law states that the ratio of the sine of an angle to the length of the side opposite that angle is constant in any triangle.
Let's use the law of sines to find the angle:
sin(B) / b = sin(A) / a
where:
B is the angle we want to find
sin(B) is the sine of the angle B
b is the magnitude of the force of 60 Newtons (60 N)
A is the included angle (30 degrees)
a is the magnitude of the resultant force (40.87 N)
Plugging in the values:
sin(B) / 60 = sin(30) / 40.87
sin(B) = (60 / 40.87) * sin(30)
sin(B) = 1.46826 * 0.5
sin(B) ≈ 0.73413
Now, we can take the inverse sine to find the angle B:
B ≈ sin^(-1)(0.73413)
B ≈ 48.99 degrees
Therefore, to the nearest degree, the angle that the resultant force makes with the force of 60 Newtons is approximately 49 degrees.