Two tugboats pull a disabled supertanker. Each tug exerts a constant force of $1.80\times {10}^6$ N, one $14°$ west of north and the other $14°$ east of north, as they pull the tanker $0.75$ km toward the north. What is the total work they do on the supertanker?

Verified step by step guidance

1. Step 1: Understand the problem. The work done by a force is calculated using the formula: $W=Fdcos\left(\theta \right)$, where $W$ is the work, $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and the direction of displacement. Here, two forces are applied at angles 14° west of north and 14° east of north, and the displacement is 0.75 km toward the north.
2. Step 2: Convert the displacement from kilometers to meters. Since 1 km = 1000 m, the displacement is $0.75\times 1000=750$ meters.
3. Step 3: Calculate the work done by each tugboat. For each tugboat, the force is $1.80\times 10^6$ N, the displacement is 750 m, and the angle between the force and the displacement is 14°. Use the formula $W=Fdcos\left(\theta \right)$ to calculate the work done by each tugboat.
4. Step 4: Add the work done by both tugboats. Since the forces are symmetric (one 14° west of north and the other 14° east of north), the total work is simply the sum of the work done by each tugboat. This is because both forces contribute positively to the displacement in the northward direction.
5. Step 5: Ensure the units are consistent. The force is in newtons (N), the displacement is in meters (m), and the angle is in degrees. The result of the work calculation will be in joules (J), as $W=Fdcos\left(\theta \right)$ inherently gives work in joules when using SI units.

Key Concepts

Work

In physics, work is defined as the product of the force applied to an object and the distance over which that force is applied, in the direction of the force. Mathematically, it is expressed as W = F × d × cos(θ), where W is work, F is the force, d is the distance, and θ is the angle between the force and the direction of motion. Understanding this concept is crucial for calculating the work done by the tugboats on the supertanker.

Vector Components

Forces acting at angles can be resolved into their vector components, typically along the x (horizontal) and y (vertical) axes. In this scenario, the tugboats exert forces at angles of 14° west and east of north, which means we need to calculate the northward (y-axis) and east/west (x-axis) components of each force. This decomposition is essential for determining the net force acting on the supertanker.

Net Force

The net force is the vector sum of all individual forces acting on an object. In this case, the net force on the supertanker is the combination of the northward components of the forces exerted by both tugboats. Calculating the net force is necessary to find the total work done, as work depends on the effective force acting in the direction of the displacement.