matthmaticaly determine the resultant force,AandB,act an object.force A is 85N and is at an angle of 20° to the horzontal,force B is125N and is angle of 60° to the horzontal.
We could use vectors
resultant = 85(cos20°, sin20°) + 125(cos 60°, sin60°)
= (78.8739, 29.0717) + (62.5, 108.2532)
= (142.3739, 137.3249)
magnitude = √(142.3739^2 + 137.3249^2) = appr 197.8 N
angle of resultant = tan^-1 (137.3249/142.3739) = 43.97..° = appr 44°
or:
make a sketch to get a triangle with sides 85 and 125 and the contained
angle of 140° (use basic geometry to get that angle)
by cosine law:
|R|^2 = 85^2 + 125^2 - 2(85)(125)cos140
|R| = 197.8 , just like before
To mathematically determine the resultant force, A and B, acting on an object, we can use vector addition. Here's how you can do it step by step:
1. Draw a diagram: Draw a diagram representing the forces A and B. Label the magnitudes and angles accordingly. In this case, draw a horizontal line to represent the horizontal direction and label it as the x-axis. Then draw a vertical line to represent the vertical direction and label it as the y-axis. Finally, draw arrows representing forces A and B, with their angles measured from the positive x-axis.
2. Resolve the forces: Resolve each force A and B into their horizontal and vertical components. To do this, determine the horizontal and vertical changes using trigonometric functions such as sine and cosine. The horizontal component (Ax or Bx) can be determined using the equation: Ax = A * cos(angle) and Bx = B * cos(angle). The vertical component (Ay or By) can be determined using the equation: Ay = A * sin(angle) and By = B * sin(angle), where angle is in radians.
3. Calculate the resultant force in the x-direction: Add the horizontal components of the forces A and B together to find the resultant force in the x-direction. Use the equation: Rx = Ax + Bx.
4. Calculate the resultant force in the y-direction: Add the vertical components of the forces A and B together to find the resultant force in the y-direction. Use the equation: Ry = Ay + By.
5. Find the magnitude and direction of the resultant force: Use the resultant forces in the x and y directions (Rx and Ry) to calculate the magnitude of the resultant force (R) using the Pythagorean theorem: R = sqrt(Rx^2 + Ry^2). The direction of the resultant force (θ) can be calculated using the inverse tangent function: θ = atan(Ry/Rx).
6. Substitute the values: Substitute the given values into the equations, such as A = 85N, B = 125N, angle of A = 20°, and angle of B = 60°. Convert the angles to radians if necessary.
7. Calculate the components: Calculate Ax, Ay, Bx, and By using the formulas mentioned in step 2.
8. Calculate the resultant force: Calculate Rx and Ry using the formulas mentioned in step 3.
9. Calculate the magnitude and direction: Calculate R using the formula mentioned in step 5. Also, calculate θ using the inverse tangent function.
By following these steps, you will be able to mathematically determine the resultant force for forces A and B acting on an object.