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Chapter 10: Problem 18

\(\cdot\) What is the power output in horsepower of an electric motor turning at 4800 rev/min and developing a torque of 4.30 \(\mathrm{N} \cdot \mathrm{m}\) ?

Short Answer

Expert verified
The motor's power output is approximately 2.90 horsepower.

Step by step solution

01

Understanding Power in Mechanical Systems

First, understand that power in a mechanical system can be calculated using the formula \( P = \tau \cdot \omega \), where \( \tau \) is the torque and \( \omega \) is the angular velocity. The torque is given as 4.30 \( \mathrm{N} \cdot \mathrm{m} \).
02

Convert RPM to Radians per Second

To find the angular velocity \( \omega \) in radians per second, convert 4800 revolutions per minute (rev/min) using the relation \( 1 \text{ rev} = 2\pi \text{ radians} \) and \( 1 \text{ minute} = 60 \text{ seconds} \). Thus, \( \omega = 4800 \times \frac{2\pi}{60} \).
03

Calculate Angular Velocity

Calculate \( \omega \) as follows: \( \omega = 4800 \times \frac{2 \pi}{60} = 4800 \times \frac{\pi}{30} \approx 502.65 \text{ rad/s} \).
04

Calculate Power in Watts

Substitute the values of torque and angular velocity into the power formula: \( P = 4.30 \times 502.65 = 2161.395 \text{ W} \).
05

Convert Watts to Horsepower

To convert the power from watts to horsepower, use the conversion factor \( 1 \text{ horsepower} = 745.7 \text{ W} \). Calculate the horsepower as \( \frac{2161.395}{745.7} \approx 2.90 \text{ hp} \).

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Key Concepts

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

Angular Velocity
Angular velocity is a measure of how fast something moves around a circular path. It is commonly represented by the symbol \( \omega \) and typically measured in radians per second (rad/s).

If you think of a car's steering wheel, angular velocity would represent how quickly the wheel is turned. It's a vital concept in determining the speed of rotating objects.

When dealing with mechanical systems, the formula for power, \( P = \tau \cdot \omega \), shows the role of angular velocity in calculating how much work a system can do. Higher angular velocity means more rotations are completed in a given time, directly impacting the system's power output.
Torque
Torque is a force that tends to cause rotation. You can think of it as the twist applied to an object. It鈥檚 generally measured in Newton-meters (\( \mathrm{N} \cdot \mathrm{m} \)).

The equation \( P = \tau \cdot \omega \) highlights torque as a crucial factor in mechanical power calculations. Here, \( \tau \), the torque, determines how effectively the engine can cause rotation.

For example, when you use a wrench, applying more torque will turn the bolt more effectively. In engines and machinery, torque is a critical aspect for understanding performance, especially when combined with angular velocity.
Horsepower Conversion
Horsepower is a unit often used in the context of engine power. It provides an intuitive grasp of the engine's capability compared to watts.

To convert from watts to horsepower, you use the conversion factor: \( 1 \text{ horsepower} = 745.7 \text{ watts} \). This conversion helps in assessing the performance of engines, especially in automotive contexts where horsepower is commonly referenced.

Understanding horsepower can make it easier to compare different engines, as it gives a clear picture of the power output. It also provides context for various applications, like car engines, where horsepower remains a standard unit of measure.
Revolutions per Minute to Radians per Second Conversion
When analyzing rotating systems, it is often necessary to convert between revolutions per minute (RPM) and radians per second (rad/s) to work within the International System of Units (SI).

To convert RPM to rad/s, you use the relation: \( 1 \text{ revolution} = 2\pi \text{ radians} \) and the fact that there are 60 seconds in a minute. The formula becomes:

\[ \omega = \text{RPM} \times \frac{2\pi}{60} \]

This conversion ensures consistency in calculations involving angular velocity and is essential for solving problems in mechanical engineering, such as determining the output power based on rotational speed.

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Most popular questions from this chapter

\(\bullet\) Two people carry a heavy electric motor by placing it on a light board 2.00 m long. One person lifts at one end with a force of \(400.0 \mathrm{N},\) and the other lifts at the opposite end with a force of 600.0 \(\mathrm{N}\) . (a) Start by making a free-body diagram of the motor. (b) What is the weight of the motor? (c) Where along the board is its center of gravity located?

A hollow spherical shell with mass 2.00 \(\mathrm{kg}\) rolls without slipping down a \(38.0^{\circ}\) slope. (a) Find the acceleration of the shell and the friction force on it. Is the friction kinetic or static friction? Why? (b) How would your answers to part (a) change if the mass were doubled to 4.00 \(\mathrm{kg} ?\)

\(\cdot\) The spinning figure skater. The outstretched hands and arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. (See Figure \(10.53 .\) ) When the skater's hands and arms are brought in and wrapped around his body to execute the spin, the hands and arms can be considered a thin-walled hollow cylinder. His hands and arms have a combined mass of 8.0 \(\mathrm{kg}\) . When outstretched, they span 1.8 \(\mathrm{m}\) ; when wrapped, they form a cylinder of radius 25 \(\mathrm{cm} .\) The moment of inertia about the axis of rotation of the remainder of his body is constant and equal to 0.40 \(\mathrm{kg} \cdot \mathrm{m}^{2} .\) If the skater's original angular speed is 0.40 \(\mathrm{rev} / \mathrm{s}\) what is his final angular speed?

\bullet Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly \(10^{14}\) times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid spiere, both before and after the collapse. The star's initial radius was \(7.0 \times 10^{5} \mathrm{km}\) (comparable to our sun); its final radius is 16 \(\mathrm{km}\) . If the original star rotated once in 30 days, find the angular speed of the neutron star.

\(\bullet\) A 750 gram grinding wheel 25.0 \(\mathrm{cm}\) in diameter is in the shape of a uniform solid disk. (We can ignore the small hole at the center.) When it is in use, it turns at a constant 220 \(\mathrm{rpm}\) about an axle perpendicular to its face through its center. When the power switch is turned off, you observe that the wheel stops in 45.0 s with constant angular acceleration due to friction at the axle. What torque does friction exert while this wheel is slowing down?
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