Problem 6 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

The total work done is approximately 2.62 \(\times 10^9\) J.

Step 1: Identify the Forces

Each tugboat exerts a force of \(1.80 \times 10^6\, \text{N}\). The directions are given as 14° west of north and 14° east of north. We are required to find the component of these forces in the direction of displacement, which is toward the north.

Step 2: Calculate Components of Forces

For each tugboat, the northern force component can be calculated as \(F_n = 1.80 \times 10^6 \times \cos(14°)\). Calculate this component.

Step 3: Sum the Force Components

Since the tugboats are pulling symmetrically and the east-west components cancel each other, sum the northern components: \(2 \times F_n\).

Step 4: Calculate Work Done

Convert displacement into meters: \(0.75 \text{ km} = 750 \text{ m}\). The work done is given by \(W = F_{\text{net}} \times d\), where \(F_{\text{net}}\) is the total force in the direction of displacement. Substitute and calculate the work.

Key Concepts

Force Components

In physics, forces come with both a magnitude and a direction. Sometimes, we need to break these forces into components along different directions. This is especially helpful when forces act at an angle, just like in our tugboats example. For each tug, the actual force is exerted at 14° from the north.

To find out how much of this force is pulling directly north, we use the concept of force components.

• Northern Component: The northward pull is found using trigonometric functions. To get the northern component, we multiply the force by the cosine of the angle:

\[ F_n = F imes \cos(14°) \] Where \( F \) is the total force exerted by the tugboat. This tells us exactly how much of the force is pulling in the direction of displacement, which is north in this case.

Vector Addition

Adding forces together isn't always straightforward. When forces are vector quantities, we need to consider both magnitude and direction. In our example, we have two forces from the tugboats pulling the tanker.

• Both forces have components that pull toward the north, and these components add directly.
• East-west components cancel because of symmetrical arrangement around the north direction.

By adding the northward components of each force, we can find the total northward force acting on the tanker.
Vector addition allows us to find out how much force actually impacts the motion of the object in the desired direction.

Displacement

Displacement refers to how far an object moves in a certain direction. Unlike distance, displacement is a vector quantity which means it has both magnitude and direction. In this problem, the tanker is being displaced towards the north by 0.75 km.

• We first convert this displacement to meters because that's the standard unit in physics calculations:

\[ 0.75 ext{ km} = 750 ext{ m} \]
Displacement is not just how far something moves, but where it moves to, highlighting the importance of direction in vector problems.

Net Force

When multiple forces act on an object, it's their combined effect that determines the object's motion. This combined force is known as the net force. In this problem, each tugboat applies a force, but since they pull in slightly different directions, we only look at the northern component of each.

• Using vector addition, we sum the northward components of the two forces. This resultant force in the northern direction is our net force:

\[ F_{\text{net}} = 2 imes F_n \]
The net force allows us to calculate the work done on the supertanker, which is given as the product of net force and displacement. Understanding net force is crucial in analyzing any situation where multiple forces interact.