An object is situated so that its center of mass is located at the origin.Three forces X,Y, and Z act on the object at the same time. Force X passes through the point (-3, -4), forve Y passes through the point (5,-12), and force Z passes throught the point (15,8). Forces X,Y, and Z have magnitures of 80,91, and 51 Newtons respectively. Make a sketch of the situation then find the magnitude and direction of the resultant force acting on the object.
X: 80*(-3i-4j)/5 = -48i - 64j
Y: 91*(5i-12j)/13 = 35i - 84j
Z: 51*(15i+8j)/17 = 45i + 24j
sum: 32i -124j = 128N at -75.5°
128N
To find the magnitude and direction of the resultant force acting on the object, we can use vector addition. Here's how you can do it:
1. Sketch the Situation:
Draw a coordinate system with the center of mass at the origin (0,0). Then, draw vectors X, Y, and Z from their respective points of application to the origin.
```
|
Z | Y
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\|/
-----------O------------ X
|
```
Note: The lengths of the vectors are not shown to scale.
2. Calculate the Resultant Vector:
To find the resultant vector, add the three forces vectorially. Start by finding the components of each force:
- Force X: (-3, -4) with a magnitude of 80 N
X component: 80 * cos(θ1) = 80 * cos(arctan((-4)/(-3)))
Y component: 80 * sin(θ1) = 80 * sin(arctan((-4)/(-3)))
- Force Y: (5, -12) with a magnitude of 91 N
X component: 91 * cos(θ2) = 91 * cos(arctan((-12)/5))
Y component: 91 * sin(θ2) = 91 * sin(arctan((-12)/5))
- Force Z: (15, 8) with a magnitude of 51 N
X component: 51 * cos(θ3) = 51 * cos(arctan(8/15))
Y component: 51 * sin(θ3) = 51 * sin(arctan(8/15))
3. Add the Components:
Sum up the X and Y components of the three forces:
X component of the resultant = X component of X + X component of Y + X component of Z
Y component of the resultant = Y component of X + Y component of Y + Y component of Z
4. Find the Magnitude and Direction:
Use the Pythagorean theorem and inverse tangent function to find the magnitude and direction of the resultant:
Resultant magnitude = sqrt((X component of the resultant)^2 + (Y component of the resultant)^2)
Resultant direction = arctan((Y component of the resultant) / (X component of the resultant))
5. Calculate the Values:
By plugging in the values from step 3 into the magnitude and direction formulas, you can calculate the final answers.
Once you obtain these values, you will have the magnitude and direction of the resultant force acting on the object.