An object is acted upon by two forces, u and v, with direction angles €u and€v, respectively. Find the direction and magnitude of the resultant force. U= 30lb €u = 0degrees v= 90lb €v =60 degrees
To find the direction and magnitude of the resultant force, we can use the concept of vector addition:
1. Start by resolving each force into horizontal and vertical components using trigonometry.
- For force u: u × cos(€u) gives the horizontal component, and u × sin(€u) gives the vertical component.
- For force v: v × cos(€v) gives the horizontal component, and v × sin(€v) gives the vertical component.
2. Add the horizontal components of the forces together to get the horizontal component of the resultant force.
R_x = (horizontal component of u) + (horizontal component of v)
3. Add the vertical components of the forces together to get the vertical component of the resultant force.
R_y = (vertical component of u) + (vertical component of v)
4. The magnitude of the resultant force, R, can be found using the Pythagorean theorem:
R = √(R_x^2 + R_y^2)
5. Finally, find the direction of the resultant force using trigonometry:
The direction angle, €R, can be found using the inverse tangent function:
€R = atan(R_y / R_x)
Now, let's calculate the resultant force:
Given:
u = 30 lb, €u = 0 degrees
v = 90 lb, €v = 60 degrees
1. Resolving forces into components:
u_x = u × cos(€u) = 30 lb × cos(0°) = 30 lb
u_y = u × sin(€u) = 30 lb × sin(0°) = 0 lb
v_x = v × cos(€v) = 90 lb × cos(60°) = 45 lb
v_y = v × sin(€v) = 90 lb × sin(60°) = 77.94 lb
2. Adding the horizontal components:
R_x = u_x + v_x = 30 lb + 45 lb = 75 lb
3. Adding the vertical components:
R_y = u_y + v_y = 0 lb + 77.94 lb = 77.94 lb
4. Calculating the magnitude of the resultant force:
R = √(R_x^2 + R_y^2) = √(75 lb^2 + 77.94 lb^2) ≈ 109.82 lb
5. Calculating the direction angle of the resultant force:
€R = atan(R_y / R_x) = atan(77.94 lb / 75 lb) ≈ 46.8 degrees
Therefore, the direction of the resultant force is approximately 46.8 degrees, and its magnitude is approximately 109.82 lb.