Three forces of magnitude 15N, 10N and 5N act on a particle in the direction which make 120° with one another. Find the resultant and the angle the resultant makes with the x-axis?
Fr = 15N[0o] + 10N.[120o] + 5N.[240o]. = Resultant force.
X = 15*Cos0 + 10*Cos120 + 5*Cos240 = 7.5 N.
Y = 15*sin0 + 10*sin120 + 5*sin240 = 4.33 N.
Fr = 7.5 + 4.33i = 8.66N.[30o] CCW. = Resultant force.
15cis0° + 10cis120° + 5cis240° = 5√3 cis30°
To find the resultant of the three forces, we can use the method of vector addition. We will find the horizontal and vertical components of each force and then add these components to find the resultant.
Let's break down the forces into their horizontal and vertical components:
Force 1 (15N):
Horizontal component = 15N * cos(120°) = -7.5N
Vertical component = 15N * sin(120°) = 12.99N
Force 2 (10N):
Horizontal component = 10N * cos(120°) = -5N
Vertical component = 10N * sin(120°) = 8.66N
Force 3 (5N):
Horizontal component = 5N * cos(120°) = -2.5N
Vertical component = 5N * sin(120°) = 4.33N
Now, let's add the horizontal and vertical components separately:
Horizontal component of the resultant = (-7.5N) + (-5N) + (-2.5N) = -15N
Vertical component of the resultant = (12.99N) + (8.66N) + (4.33N) = 25.98N
To find the magnitude of the resultant force, we use the Pythagorean theorem:
Resultant magnitude = sqrt((horizontal component)^2 + (vertical component)^2)
= sqrt((-15N)^2 + (25.98N)^2)
= sqrt(225N^2 + 675N^2)
= sqrt(900N^2)
= 30N
Now, let's find the angle that the resultant makes with the x-axis using the inverse tangent function:
Angle = atan(vertical component / horizontal component)
= atan(25.98N / -15N)
= atan(-1.732)
≈ -60°
Therefore, the magnitude of the resultant force is 30N, and the angle it makes with the x-axis is approximately -60°.
To find the resultant of the forces, you need to use the concept of vector addition. Here's how you can solve this problem:
1. Start by representing the three forces as vectors. Let's call the vectors F1, F2, and F3. We'll consider F1 = 15N, F2 = 10N, and F3 = 5N.
2. Since the forces are making an angle of 120° with each other, we need to find their components along the x and y axes.
- For F1: The x-component (F1x) can be found using the formula F1 * cos(120°), and the y-component (F1y) can be found using F1 * sin(120°).
- For F2: The x-component (F2x) can be found using the formula F2 * cos(120° + 120°), as this force is rotated by an additional 120° relative to F1. The y-component (F2y) can be found using F2 * sin(120° + 120°).
- For F3: The x-component (F3x) can be found using the formula F3 * cos(120° + 120° + 120°), as this force is rotated by an additional 120° relative to F1 and F2. The y-component (F3y) can be found using F3 * sin(120° + 120° + 120°).
3. Calculate the resultant components by adding the corresponding x and y components of the forces:
- Resultant x-component (Rx) = F1x + F2x + F3x
- Resultant y-component (Ry) = F1y + F2y + F3y
4. Use the Pythagorean theorem to find the magnitude of the resultant force (R):
- R = sqrt(Rx^2 + Ry^2)
5. Finally, calculate the angle (θ) between the resultant force and the x-axis using the arctangent function:
- θ = arctan(Ry / Rx)
Now let's plug in the numbers and calculate the solution:
F1x = 15N * cos(120°)
F1y = 15N * sin(120°)
F2x = 10N * cos(240°)
F2y = 10N * sin(240°)
F3x = 5N * cos(360°)
F3y = 5N * sin(360°)
Rx = F1x + F2x + F3x
Ry = F1y + F2y + F3y
R = sqrt(Rx^2 + Ry^2)
θ = arctan(Ry / Rx)