Tugboat resultant force and equilibrant direction

Key takeaway: Tugboat force addition uses vector components, bearings, and the equilibrant as the same magnitude pointing in the opposite direction.

Interpreting the two tugboat forces

The forces 1500 N and 1800 N are not acting along the same line, so they must be combined as vectors rather than ordinary numbers. The problem gives bearings relative to straight ahead, which means direction matters as much as size. To solve it correctly, separate each force into components, add those components, and then convert the result back into a magnitude and direction.

The given answer check, 2708 N at 98.6°, is useful for verifying the final result, but the full method still matters. The equilibrant will have the same magnitude as the resultant force and point exactly opposite to it.

Resolving the forces into components

For the tugboat vector components, choose one axis straight ahead and another axis to the side. Then write each tug force as horizontal and vertical parts using trigonometry. Since the angles are given by bearings to the left and right of straight ahead, the signs of the side components must reflect the direction each tug pulls.

After resolving both forces, add the forward components together and add the sideways components together. The combined vector gives the net force acting on the freighter. Once the magnitude and direction are known, you can describe the motion of the freighter in a single resulting direction.

Resultant force and equilibrant

Using vector addition, the resultant force is 2708 N at a direction of 98.6°. This means the freighter moves in the direction of the combined pull, not in the direction of either tug alone. The angle should be interpreted according to the bearing convention used in your class or textbook, so label your diagram carefully.

The equilibrant has the same magnitude as the resultant force but points in the opposite direction. So if the resultant is 2708 N at 98.6°, the equilibrant is also 2708 N but directed 180° opposite. In words, the equilibrant is the single force that would balance the tugboats and keep the freighter from accelerating.

Why the method works

This problem relies on the fact that forces are vectors, not scalars. A force of 1500 N does not simply add to 1800 N to give 3300 N, because the directions are different. The geometry of the vector triangle is what creates the final magnitude and angle.

It is helpful to draw the two tugboat forces tail-to-tail or head-to-tail before calculating. The sketch shows why one component adds while another subtracts. Once the geometry is clear, the trigonometry becomes much easier to manage, and the equilibrant is just the exact opposite of the resultant.

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Pitfalls the pros know 👇 The easiest way to lose marks in this tugboat force vector question is to add 1500 N and 1800 N directly. That only works if the forces point in the same direction, which they do not here. Another common mistake is mixing up the bearing directions for left and right of straight ahead, which can flip the side components and change the final angle.

A second trap is forgetting that the equilibrant is not a new calculation from scratch. It always has the same size as the resultant and points in the opposite direction. If the resultant direction is written incorrectly, the equilibrant will also be wrong. The safest habit is to draw the component diagram first, label the axes, and then check whether your final answer matches the given 2708 N, 98.6° benchmark.

What if the problem changes? If the first tugboat pulls with 1200 N instead of 1500 N while the second tugboat remains at 1800 N, the same vector method applies but the resultant force will change. You would still split both forces into components, add the forward and sideways parts, and then compute the new magnitude and direction. A smaller first force would generally reduce the size of the resultant and may shift the direction closer to the second tugboat's bearing.

Another variation is to keep the forces the same but change one bearing, for example making the second tugboat pull 140° to the right of straight ahead instead of 130°. That small angle change can noticeably alter the sideways component and therefore the final direction of the freighter. These variants show why vector problems depend on both magnitude and direction, not just the numbers alone.

Tags: vector resolution of forces, equilibrant force, bearing angle notation