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Chapter 8: Problem 67

(II) An outboard motor for a boat is rated at 55 hp. If it can move a particular boat at a steady speed of \(35 \mathrm{~km} / \mathrm{h}\), what is the total force resisting the motion of the boat?

Short Answer

Expert verified
The force resisting the boat's motion is approximately 4219.55 N.

Step by step solution

01

Convert Horsepower to Watts

To calculate the force, we need to first convert the motor's power rating from horsepower to watts. 1 horsepower is equivalent to approximately 745.7 watts. So, for a 55 hp motor, the power in watts is calculated as follows: \( 55 \times 745.7 = 41013.5 \) watts.
02

Convert Speed to Meters per Second

The boat's speed needs to be in meters per second to match the standard units for calculations involving power and force. Since 1 km/h is equivalent to \( \frac{1000}{3600} \) m/s, the boat's speed of 35 km/h can be converted as follows: \( 35 \times \frac{1000}{3600} \approx 9.72 \) m/s.
03

Apply Power Formula to Find Force

We use the formula relating power \( P \), force \( F \), and velocity \( v \): \( P = F \times v \). Rearranging to solve for force gives us \( F = \frac{P}{v} \). Substituting the known values, \( F = \frac{41013.5}{9.72} \approx 4219.55 \text{ Newtons} \).

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Key Concepts

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

Horsepower Conversion
Horsepower is a common unit of power often used to measure the power output of engines. When solving problems in physics, it's crucial to first convert horsepower into a more widely used unit, such as watts. This ensures mathematical consistency and ease of calculation.

Here's how you can perform the conversion:
  • Remember: 1 horsepower (hp) equals approximately 745.7 watts.
  • To convert horsepower to watts, multiply the horsepower value by 745.7.
For example, a motor rated at 55 horsepower would be converted to watts by calculating:\[55 \times 745.7 = 41013.5 \text{ watts}\]It's essential to understand this conversion as many equations in physics, particularly those involving power, rely on the use of watts.
Power and Force Relationship
The relationship between power, force, and velocity is a fundamental concept in physics. It's encapsulated in the formula:\[P = F \times v\]where:
  • \( P \) is the power in watts,
  • \( F \) is the force in newtons, and
  • \( v \) is the velocity in meters per second.

To solve for the force resisting the motion of an object, you can rearrange the formula as follows:\[F = \frac{P}{v}\]This equation tells us that the force is equal to the power divided by the velocity.
For example, if a boat's motor generates 41013.5 watts of power and moves at a speed of 9.72 m/s, then the force is:\[F = \frac{41013.5}{9.72} \approx 4219.55 \text{ Newtons}\]This straightforward relationship helps in understanding how power, force, and motion interconnect in real-world situations.
Unit Conversion in Physics
Unit conversion is a pivotal skill in physics that enables the transformation of various measurements into compatible units for calculation. This is especially important when working with equations involving different physical quantities.

In our example, we needed to convert speed from kilometers per hour to meters per second to align with the units of power and force:
  • 1 kilometer per hour (km/h) is equivalent to \( \frac{1000}{3600} \) meters per second (m/s).
  • So, to convert 35 km/h into m/s: \[35 \times \frac{1000}{3600} \approx 9.72 \text{ m/s}\]
Converting units consistently ensures that calculations are accurate, which prevents errors that could arise from using mismatched units. It鈥檚 a fundamental process that underpins problem-solving across all areas of physics.

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Most popular questions from this chapter

(I) If a car generates 18 hp when traveling at a steady \(95 \mathrm{~km} / \mathrm{h},\) what must be the average force exerted on the car due to friction and air resistance?

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir. To store the energy produced in 1.0 hour by a 180 -MW electric power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is \(380 \mathrm{~m}\) above the lower one, and we can neglect the small change in depths of each. Water has a mass of \(1.00 \times 10^{3} \mathrm{~kg}\) for every \(1.0 \mathrm{~m}^{3}\)

(II) A ski area claims that its lifts can move 47,000 people per hour. If the average lift carries people about \(200 \mathrm{~m}\) (vertically) higher, estimate the maximum total power needed.

(III) The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or \(F(r)=F_{0}\left[2(\sigma / r)^{13}-(\sigma / r)^{7}\right],\) where \(r\) is the separation between the two atoms, and \(\sigma\) and \(F_{0}\) are constant. For an oxygen molecule (which is diatomic) \(F_{0}=9.60 \times 10^{-11} \mathrm{~N}\) and \(\sigma=3.50 \times 10^{-11} \mathrm{~m} .\) (a) Integrate the equation for \(F(r)\) to determine the potential energy \(U(r)\) of the oxygen molecule. (b) Find the equilibrium distance \(r_{0}\) between the two atoms ( \(c\) ) Graph \(F(r)\) and \(U(r)\) between \(0.9 r_{0}\) and \(2.5 r_{0}\)

A particle of mass \(m\) moves under the influence of a potential energy $$ U(x)=\frac{a}{x}+b x $$ where \(a\) and \(b\) are positive constants and the particle is restricted to the region \(x>0\). Find a point of equilibrium for the particle and demonstrate that it is stable.
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