Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

(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?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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, \( F = \frac{41013.5}{9.72} \approx 4219.55 \text{ Newtons} \).

Key Concepts

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’s a fundamental process that underpins problem-solving across all areas of physics.