Question

An object is suspended by two cables attached at a single point. The force applied on one cable has a magnitude of 125 pounds and acts at an angle of 37°. The force on the other cable is 75 pounds at an angle of 150°. Part A: Write each vector in component form. Show all necessary work. Part B: Find the dot product of the vectors. Show all necessary calculations Part C: Use the dot product to find the angle between the cables. Round the answer to the nearest degree. Show all necessary calculations.

Answer 1

The problem requires finding the vector components using trigonometry, calculating the dot product, and then determining the angle between the cables by using the dot product and magnitudes of the vectors.

Step-by-step explanation:

The student question involves calculating the vector components of forces, finding the dot product of two vectors, and using this to determine the angle between cables. This is a multi-part physics problem involving vector analysis.

Part A: Vector Components

To express each vector in component form, we use trigonometry. For the first cable force (F1) of 125 pounds at an angle of 37°, the components are:

F1x = F1 * cos(37°) = 125 * cos(37°)

F1y = F1 * sin(37°) = 125 * sin(37°)

For the second cable force (F2) of 75 pounds at an angle of 150°, the components are:

F2x = F2 * cos(150°) = 75 * cos(150°)

F2y = F2 * sin(150°) = 75 * sin(150°)

Part B: Dot Product Calculation

The dot product of the two vectors is calculated as:

Dot Product = F1x * F2x + F1y * F2y

Part C: Angle Between Cables

To find the angle between the cables, we use the dot product and magnitudes of the vectors:

cos(θ) = (Dot Product) / (Magnitude of F1 * Magnitude of F2)

Then calculate the angle θ using the inverse cosine function and round to the nearest degree.