Two tugboats pull a large ship back to the harbor.

One tugboat is pulling the ship west with a force of 3700 lbs, and the other is pulling the ship 10° north of west with a force of 4200 lbs.

What is the direction and magnitude of the resultant force?

Question

High School

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Answer :

onyebuchinnaji onyebuchinnaji · Answer 1 · 2023-09-28T02:24:04-04:00

The direction and magnitude of the resultant force is 7,870.1 lbs at 5.3 ° south of west.

We can start by breaking down the forces into their x and y components:

The force of 3700 lbs pulling west has an x-component of -3700 lbs and a y-component of 0 lbs.

The force of 4200 lbs pulling 10° north of west has an x-component of -4200 cos(10°) lbs and a y-component of 4200 sin(10°) lbs.

The x-component of the resultant force is;

Fx = -3700 lbs - 4200 cos(10°) lbs

Fx = -7,836.19 lbs

The y-component of the resultant force is;

Fy = 4200 sin(10°) lbs

Fy = 729.32 lbs

To find the magnitude and direction of the resultant force, we can use the Pythagorean theorem and inverse tangent function:

The magnitude of the resultant force is;

F = √(Fx² + Fy²)

F = √ (-7836.19² + 729.32²)

F = 7,870.1 lbs

The direction of the resultant force is;

θ = arc tan (Fy/Fx)

θ = arc tan (729.32 / 7836.19)

θ = 5.3⁰

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