Two dogs are fighting over a stuffed animal. One pulls to the right with a force of 88 N at 18° above the horizontal. The other pulls down and to the left with a force of 74 N at 12° below the horizontal. Ignoring the weight of the toy, if the total work on the animal horizontally was 45.2 J and vertically was 35.4 J, what is the magnitude of the total displacement of the animal?
To find the magnitude of the total displacement of the animal, we need to break down the given information and solve it step by step. Here's how you can approach this problem:
Step 1: Resolve Forces
Start by resolving the given forces into their horizontal and vertical components.
For the dog pulling to the right with a force of 88 N at 18° above the horizontal:
Horizontal component = 88 N * cos(18°)
Vertical component = 88 N * sin(18°)
For the dog pulling down and to the left with a force of 74 N at 12° below the horizontal:
Horizontal component = 74 N * cos(12°)
Vertical component = -74 N * sin(12°) (negative because it is down)
Step 2: Calculate Work
Next, calculate the work done by each dog in the horizontal and vertical directions:
Work done horizontally = (force pulling horizontal) * (horizontal displacement)
Work done vertically = (force pulling vertically) * (vertical displacement)
We are given that the total work done horizontally is 45.2 J and the total work done vertically is 35.4 J. So we can set up two equations:
Equation 1: Work horizontally = (force 1 horizontal) * (horizontal displacement) + (force 2 horizontal) * (horizontal displacement)
45.2 J = (88 N * cos(18°)) * (horizontal displacement) + (74 N * cos(12°)) * (horizontal displacement)
Equation 2: Work vertically = (force 1 vertical) * (vertical displacement) + (force 2 vertical) * (vertical displacement)
35.4 J = (88 N * sin(18°)) * (vertical displacement) - (74 N * sin(12°)) * (vertical displacement)
Step 3: Solve the Equations
Solve the two equations simultaneously to find the unknowns, horizontal displacement, and vertical displacement.
Step 4: Find the Total Displacement
Finally, use the Pythagorean theorem to find the magnitude of the total displacement:
Total displacement = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
By following these steps, you can find the magnitude of the total displacement of the animal.