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Chapter 6: Problem 6

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?

Short Answer

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

Step by step solution

01

Identify the Forces

Each tugboat exerts a force of \(1.80 \times 10^6\, \text{N}\). The directions are given as 14掳 west of north and 14掳 east of north. We are required to find the component of these forces in the direction of displacement, which is toward the north.
02

Calculate Components of Forces

For each tugboat, the northern force component can be calculated as \(F_n = 1.80 \times 10^6 \times \cos(14掳)\). Calculate this component.
03

Sum the Force Components

Since the tugboats are pulling symmetrically and the east-west components cancel each other, sum the northern components: \(2 \times F_n\).
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳 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:
\[ F_n = F imes \cos(14掳) \] 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.

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Most popular questions from this chapter

You are asked to design spring bumpers for the walls of a parking garage. A freely rolling 1200-kg car moving at 0.65 m/s is to compress the spring no more than 0.090 m before stopping. What should be the force constant of the spring? Assume that the spring has negligible mass.

A proton with mass 1.67 \(\times\) 10\(^{-27}\) kg is propelled at an initial speed of 3.00 \(\times\) 10\(^5\) m/s directly toward a uranium nucleus 5.00 m away. The proton is repelled by the uranium nucleus with a force of magnitude \(F = \alpha/x^2\), where \(x\) is the separation between the two objects and \(\alpha = 2.12 \times 10^{-26} \, \mathrm{N} \cdot \mathrm{m}^2\). Assume that the uranium nucleus remains at rest. (a) What is the speed of the proton when it is \(8.00 \times 10^{-10}\) m from the uranium nucleus? (b) As the proton approaches the uranium nucleus, the repulsive force slows down the proton until it comes momentarily to rest, after which the proton moves away from the uranium nucleus. How close to the uranium nucleus does the proton get? (c) What is the speed of the proton when it is again 5.00 m away from the uranium nucleus?

A 4.00-kg block of ice is placed against a horizontal spring that has force constant \(k\) = 200 N/m and is compressed 0.025 m. The spring is released and accelerates the block along a horizontal surface. Ignore friction and the mass of the spring. (a) Calculate the work done on the block by the spring during the motion of the block from its initial position to where the spring has returned to its uncompressed length. (b) What is the speed of the block after it leaves the spring?

Two blocks are connected by a very light string passing over a massless and frictionless pulley (\(\textbf{Fig. E6.7}\)). Traveling at constant speed, the 20.0-N block moves 75.0 cm to the right and the 12.0-N block moves 75.0 cm downward. How much work is done (a) on the 12.0-N block by (i) gravity and (ii) the tension in the string? (b) How much work is done on the 20.0-N block by (i) gravity, (ii) the tension in the string, (iii) friction, and (iv) the normal force? (c) Find the total work done on each block.

About 50,000 years ago, a meteor crashed into the earth near present-day Flagstaff, Arizona. Measurements from 2005 estimate that this meteor had a mass of about 1.4 \(\times\) 10\(^8\) kg (around 150,000 tons) and hit the ground at a speed of 12 km/s. (a) How much kinetic energy did this meteor deliver to the ground? (b) How does this energy compare to the energy released by a 1.0-megaton nuclear bomb? (A megaton bomb releases the same amount of energy as a million tons of TNT, and 1.0 ton of TNT releases 4.184 \(\times\) 10\(^9\) J of energy.)
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