/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 The motor of a ski boat generate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 61

The motor of a ski boat generates an average power of \(7.50 \times 10^{4} \mathrm{~W}\) when the boat is moving at a constant speed of \(12 \mathrm{~m} / \mathrm{s}\). When the boat is pulling a skier at the same speed, the engine must generate an average power of \(8.30 \times 10^{4} \mathrm{~W}\). What is the tension in the tow rope that is pulling the skier?

Short Answer

Expert verified
The tension in the tow rope is approximately 667 N.

Step by step solution

01

Understanding the Given Values

The power generated by the motor when the boat is moving alone is given as \(7.50 \times 10^4\, \text{W}\), and when it is pulling a skier, it is \(8.30 \times 10^4\, \text{W}\). The boat's speed (which is constant) is \(12\, \text{m/s}\). We need to find the tension in the rope when the skier is being pulled.
02

Calculate the Additional Power Needed

First, identify the additional power required to pull the skier by subtracting the power needed for the boat alone from the power needed when pulling the skier:\[\Delta P = (8.30 \times 10^4) - (7.50 \times 10^4) = 8.0 \times 10^3\, \text{W}.\]
03

Relate Additional Power with Force/Tension

The formula for power in terms of force and velocity is given by:\[P = F \cdot v,\]where \(P\) is power, \(F\) is force (or tension in this case), and \(v\) is velocity. We rearrange this to solve for force:\[F = \frac{P}{v}.\]
04

Solve for Tension in the Tow Rope

Substitute the known values into the equation for force to find the tension:\[F = \frac{8.0 \times 10^3\, \text{W}}{12\, \text{m/s}} = 666.67\, \text{N}.\]
05

Conclusion

The tension in the tow rope pulling the skier, while the boat is moving at a constant speed of \(12\, \text{m/s}\), is approximately \(667\, \text{N}\) (rounded to three significant figures).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Power
In physics, power is a crucial concept that translates to the rate at which work is done or energy is transferred. When a ski boat motor is running, it is essentially doing work to move the boat through water by expending energy at a certain rate. Power is measured in watts (W), where 1 watt is equal to 1 joule per second.
  • The motor's power output while moving alone was given as \(7.50 \times 10^4\, \text{W}\).
  • When the motor pulls a skier, the power jumps to \(8.30 \times 10^4\, \text{W}\).
This difference in power highlights the extra energy required to overcome the additional resistance generated by the skier. Understanding power helps in realizing how engineers design engines better to maximize efficiency and handle additional loads without compromising speed or performance.
Force
Force is what changes or tends to change the state of rest or motion of an object. It's a vector quantity, meaning it has both magnitude and direction. In the context of our ski boat problem, force is applied forward by the boat's motor to maintain its movement across the water.
Force is measured in newtons (N), and it can be calculated using the formula:
  • When there's a constant velocity, the net force is balanced by the forces of resistance like drag.
  • Additional power supplied indicates additional force to overcome this resistance.
In our problem, by knowing the extra power when pulling the skier, we deduce the extra force (tension) needed, which is calculated to be 667 N.
Tension
Tension in physics often relates to the force experienced by a rope or cable when it's used to transmit a force. In our scenario, tension is the force exerted by the tow rope on the skier.
  • Tension is created when the rope pulls on the skier with a force that matches the resistance opposing the skier's movement.
  • The calculated tension of 667 N means the tow rope successfully transmits this force to pull the skier along at the same speed as the boat.
The tension results from the need for additional power beyond what is required to move the boat alone, and it's critical for ensuring the skier moves smoothly without interrupting the boat's speed.
Velocity
Velocity is a measure of how fast an object moves in a particular direction. Unlike speed, which only considers how fast something is moving, velocity adds the component of direction, making it a vector quantity.
In the scenario of the ski boat:
  • The constant speed of 12 m/s dictates that velocity remains unchanged in both magnitude and direction.
  • This constant velocity ensures that the skier is pulled at an even pace, making it comfortable and safe.
This constancy also provides a reliable basis for calculating the required forces and power because changes in velocity would complicate the calculations and introduce additional forces into consideration, like acceleration.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A \(5.00 \times 10^{2}-\mathrm{kg}\) hot-air balloon takes off from rest at the surface of the earth. The nonconservative wind and lift forces take the balloon up, doing \(+9.70 \times 10^{4} \mathrm{~J}\) of work on the balloon in the process. At what height above the surface of the earth does the balloon have a speed of \(8.00 \mathrm{~m} / \mathrm{s}\) ?

A roller coaster \((375 \mathrm{~kg})\) moves from \(A(5.00 \mathrm{~m}\) above the ground) to \(B(20.0 \mathrm{~m}\) above the ground). Two nonconservative forces are present: friction does \(-2.00 \times 10^{4} \mathrm{~J}\) of work on the car, and a chain mechanism does \(+3.00 \times 10^{4} \mathrm{~J}\) of work to help the car up a long climb. What is the change in the car's kinetic energy, \(\Delta \mathrm{KE}=\mathrm{KE}_{\mathrm{f}}-\mathrm{KE}_{0}\), from \(A\) to \(B\) ?

A truck is traveling at \(11.1 \mathrm{~m} / \mathrm{s}\) down a hill when the brakes on all four wheels lock. The hill makes an angle of \(15.0^{\circ}\) with respect to the horizontal. The coefficient of kinetic friction between the tires and the road is \(0.750\). How far does the truck skid before coming to a stop?

Interactive Solution 6.33 at presents a model for solving this problem. A slingshot fires a pebble from the top of a building at a speed of \(14.0 \mathrm{~m} / \mathrm{s}\). The building is \(31.0 \mathrm{~m}\) tall. Ignoring air resistance, find the speed with which the pebble strikes the ground when the pebble is fired (a) horizontally, (b) vertically straight up, and (c) vertically straight down.

Bicyclists in the Tour de France do enormous amounts of work during a race. For example, the average power per kilogram generated by Lance Armstrong \((m=75.0)\) is \(6.50 \mathrm{~W}\) per kilogram of his body mass. (a) How much work does he do during a \(135-\mathrm{km}\) race in which his average speed is \(12.0 \mathrm{~m} / \mathrm{s} ?\) (b) Often, the work done is expressed in nutritional Calories rather than in joules. Express the work done in part (a) in terms of nutritional Calories, noting that 1 joule \(=2.389 \times 10^{-4}\) nutritional Calories.
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Kinematics Physics

Read Explanation

Classical Mechanics

Read Explanation

Scientific Method Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Mechanics and Materials

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.