Two forces act an angle of 120degre.The bigger force is of 40N and the resultant is perpendicular to the smaller one.Find the smaller force.
draw the diagram.
You can see immediately the resultant is from the componet for Bigger in the direction of R, and that the component of Bigger in the direction of Smaller is =opposite Smaller
40cos30=Resultant
40sin30=Smaller
you can solve for smaller, and resultant from that.
To find the smaller force, we can use the concept of vector addition and trigonometry. The given information states that two forces act at an angle of 120 degrees and the resultant is perpendicular to the smaller force. Let's break down the problem step by step.
1. First, draw a diagram representing the given information. Assume that the smaller force is represented by vector A and the larger force is represented by vector B. The angle between them is 120 degrees, and the resultant vector is perpendicular to vector A.
/\
/ \
/ \
A------B
\ /
\/
2. Since the resultant vector is perpendicular to vector A, it means that vector B and the resultant vector form a right-angled triangle. We can use trigonometry to solve this.
3. Let's denote the magnitude of the smaller force (vector A) as F_A. We are trying to find the value of F_A.
4. Using the given information, the magnitude of vector B is given as 40N.
5. In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the adjacent and opposite sides.
6. In our triangle, the hypotenuse is vector B with a magnitude of 40N. The side adjacent to the 120-degree angle is vector A with an unknown magnitude F_A.
7. We can use the trigonometric function cosine to relate the adjacent side and the hypotenuse. The cosine of an angle is equal to the adjacent side divided by the hypotenuse.
cos(120 degrees) = adjacent side (F_A) / hypotenuse (40N)
8. Applying the cosine formula:
cos(120 degrees) = F_A / 40N
9. Substitute the value of cos(120 degrees) which is -0.5 (since cos(120 degrees) = -0.5):
-0.5 = F_A / 40N
10. Solve for F_A by multiplying both sides of the equation by 40N:
-0.5 * 40N = F_A
-20N = F_A
11. Therefore, the smaller force (F_A) is -20N.
Note: The negative sign indicates that the smaller force acts in the opposite direction compared to the larger force. So, the smaller force has a magnitude of 20N and acts in the opposite direction as the larger force.
To solve this problem, we can use the concept of vector addition.
Let's denote the smaller force as F1 and the larger force as F2.
We know that the angle between the two forces is 120 degrees. Therefore, the angle between the resultant force (R) and the larger force (F2) is 90 degrees, since the resultant is perpendicular to the smaller force (F1).
Using the concept of vector addition, we can find the magnitude of the resultant force by using the Pythagorean theorem:
R^2 = F1^2 + F2^2
Given that F2 is 40N, we can substitute this value into the equation:
R^2 = F1^2 + (40N)^2
Now, we need to find the value of F1.
Since the angle between F1 and F2 is 120 degrees, we can use the law of cosines to relate the magnitudes of the two forces and the angle between them:
(F2)^2 = (F1)^2 + (R)^2 - 2F1R * cos(120 degrees)
Substituting the known values:
(40N)^2 = (F1)^2 + R^2 - 2F1R * cos(120 degrees)
Rearranging the equation and substituting R^2 using the previous equation:
(40N)^2 - R^2 = (F1)^2 - 2F1R * cos(120 degrees)
Substituting R^2 as F1^2 + (40N)^2:
(40N)^2 - (F1^2 + (40N)^2) = (F1)^2 - 2F1R * cos(120 degrees)
Canceling out the common terms and simplifying:
0 = (F1)^2 - 2F1R * cos(120 degrees)
Since cos(120 degrees) = -1/2:
0 = (F1)^2 + F1R
Factoring out F1:
0 = F1(F1 + R)
Since F1 cannot be zero (as it represents a force), we can solve the equation when F1 + R = 0:
F1 + R = 0
R = -F1
Since R cannot be negative (as it represents a magnitude), we can conclude that F1 + R = 0 is not a valid solution.
Therefore, F1 must be greater than zero.
Hence, the smaller force (F1) cannot be zero; it must have a positive magnitude.