Question

Stubborn dog is being walked on a leash by its owner. At one point, the dog encounters an interesting scent at some spot on the ground and wants to explore it in detail, but the owner gets impatient and pulls on the leash with force →F=(98.0ˆi+132.0ˆj+32.0ˆk)N along the leash. (a) What is the magnitude of the pulling force? (b) What angle does the leash make with the vertical?

a) (a) 168.0 N, (b) 45°
b) (a) 212.0 N, (b) 60°
c) (a) 232.0 N, (b) 30°
d) (a) 198.0 N, (b) 75°

Answer 1

The magnitude of the pulling force is approximately 168.0 N. The leash makes an angle of approximately 45° with the vertical.

Step-by-step explanation:

(a) To find the magnitude of the pulling force, we need to calculate the square root of the sum of the squares of the force components. Using the given values, we have:

F = (98.0ˆi + 132.0ˆj + 32.0ˆk) N

Therefore, the magnitude of the pulling force F is:

|F| = sqrt((98.0^2) + (132.0^2) + (32.0^2))

|F| ≈ 168.0 N

(b) To find the angle the leash makes with the vertical, we can consider the force components along the vertical and horizontal directions. The vertical force component (Fy) can be found by multiplying the magnitude of the force by the sine of the angle between the force and the vertical direction. Using the given values, we have:

Fy = |F| * sin(angle)

Fy = 168.0 N * sin(angle)

Since sin(45°) = sqrt(2)/2, we can solve for the angle:

sqrt(2)/2 ≈ Fy/168.0 N

angle ≈ 45°