Freda Pulliam and Yank Hardy are on opposite sides of a canal, pulling a barge with tow ropes. Freda exerts a force of 50 pounds at 20 degrees to the canal, and Yank pulls at an angle of 15 degrees with just enough force so that the resultant force vector is directly along the canal. Find the number of pounds with which Yank must pull and the magnitude of the resultant vector.
To find the force with which Yank must pull and the magnitude of the resultant vector, we can break down the given forces into their x and y components, and then use vector addition.
Step 1: Find the x and y components of Freda's force
To find the x and y components, we use the following trigonometric formulas:
Fx = F * cos(θ)
Fy = F * sin(θ)
where F is the force and θ is the angle.
For Freda's force:
Fx = 50 * cos(20)
≈ 47.743 pounds
Fy = 50 * sin(20)
≈ 17.167 pounds
Step 2: Find the x and y components of Yank's force
Since we know that the resultant force vector is directly along the canal, the y component of Yank's force must cancel out Freda's y component. Therefore, Fy for Yank's force is equal to -17.167 pounds.
To find Fx for Yank's force, we can calculate it using the following formula:
Fx = F * cos(θ)
We need to solve for F, the force with which Yank must pull.
Fx = F * cos(15)
Fx = -47.743 pounds
Solving for F:
F * cos(15) = -47.743
F = -47.743 / cos(15)
F ≈ -49.472 pounds
Since force cannot be negative, we take the magnitude of Yank's force as 49.472 pounds.
Step 3: Find the resultant vector
To find the resultant vector, we can add the x and y components of both forces.
Resultant Fx = Freda's Fx + Yank's Fx
= 47.743 - 47.743
= 0 pounds
Resultant Fy = Freda's Fy + Yank's Fy
= 17.167 - 17.167
= 0 pounds
The magnitude of the resultant vector is given by the formula:
Magnitude = sqrt(Resultant Fx^2 + Resultant Fy^2)
= sqrt(0^2 + 0^2)
= 0 pounds
Therefore, the number of pounds with which Yank must pull is approximately 49.472 pounds, and the magnitude of the resultant vector is 0 pounds.
To solve this problem, we can use vector addition to find the resultant force and then analyze its components to find the unknowns.
Step 1: Draw a diagram
Start by drawing a diagram to visualize the situation. Draw a canal with Freda on one side and Yank on the other side. Label the given angles, forces, and unknowns.
Step 2: Determine the x and y components of the forces
Break down each force into its x and y components. For Freda's force, the x component (F_x) is given by F * cos(angle) and the y component (F_y) is given by F * sin(angle). Similarly, for Yank's force, we need to find its x and y components.
Step 3: Find the x and y components of the resultant force
The resultant force can be found by summing up the x and y components of the individual forces.
R_x = F_x1 + F_x2
R_y = F_y1 + F_y2
Step 4: Set up equations for equilibrium along the canal
Since the resultant force is directly along the canal, its x component (R_x) should be equal to zero. This is because there is no net horizontal force acting on the barge. Equate R_x to zero and solve for the unknown.
Step 5: Solve for the unknowns
Using the equation from the previous step, solve for the unknown variable (the number of pounds with which Yank must pull). Plug in the given values for Freda's force and angle, and solve for Yank's force.
Step 6: Find the magnitude of the resultant force
Once you have the components of the resultant force, you can use the Pythagorean theorem to find its magnitude:
R = sqrt(R_x^2 + R_y^2)
By following these steps, you should be able to find the number of pounds with which Yank must pull and the magnitude of the resultant vector.