Question

The towing lines of two tugboats pulling horizontally on a barge are at an angle of 30° to each other. The tensions in the towing lines of the first and second tugboats are 3 kN and 4 kN respectively. Calculate the magnitude of the resultant force which the tugboats exert on the barge.

Answer 1

Answer: To calculate the magnitude of the resultant force exerted by the two tugboats on the barge, we can use vector addition. Since the towing lines are at an angle of 30° to each other, we can treat them as two vectors that form a triangle with the resultant force being the vector sum of the two towing forces.

Let's denote the towing force of the first tugboat as F1 = 3 kN, and the towing force of the second tugboat as F2 = 4 kN.

Using trigonometry, we can determine the horizontal and vertical components of the towing forces:

For F1:

Horizontal component: F1x = F1 * cos(30°)

Vertical component: F1y = F1 * sin(30°)

For F2:

Horizontal component: F2x = F2 * cos(30°)

Vertical component: F2y = F2 * sin(30°)

Now, we can add the horizontal and vertical components of the two towing forces separately to get the resultant force in the horizontal and vertical directions:

Horizontal component of resultant force: Fx = F1x + F2x

Vertical component of resultant force: Fy = F1y + F2y

Finally, we can use the Pythagorean theorem to calculate the magnitude of the resultant force:

Magnitude of resultant force: F = sqrt(Fx^2 + Fy^2)

Plugging in the values and calculating:

F1x = 3 kN * cos(30°) ≈ 2.598 kN

F1y = 3 kN * sin(30°) ≈ 1.5 kN

F2x = 4 kN * cos(30°) ≈ 3.464 kN

F2y = 4 kN * sin(30°) ≈ 2 kN

Fx = 2.598 kN + 3.464 kN ≈ 6.062 kN

Fy = 1.5 kN + 2 kN = 3.5 kN

F = sqrt(6.062 kN^2 + 3.5 kN^2) ≈ 6.964 kN

So, the magnitude of the resultant force exerted by the two tugboats on the barge is approximately 6.964 kN.