91Ó°ÊÓ

In mechanics, torque is essentially the measure of how much a force acting on an object causes that object to rotate. Imagine turning a wrench to tighten a bolt. The force you apply on the wrench acts at a distance from the bolt, creating a turning effect, or torque. It’s calculated as the product of the force and the distance from the rotation point (lever arm).
- **Formula**: Torque (\( \tau \)) can be expressed as\( \tau = F \times r \), where \( F \) is the force applied and \( r \) is the distance from the rotation point.
- **Units**: The units of torque are Newton-meters (\( \text{N} \cdot \text{m} \) ).
In the automobile engine example, the engine produces \( 255 ext{ m} imes ext{N} \) of torque – this means the force exerted can turn a lever arm that distance effectively. Torque is a critical concept in designing engines because it tells us about the engine's ability to perform work when rotation is involved.

Angular velocity is the rate at which an object rotates or revolves around an external point. It captures how fast something is spinning and is particularly used in contexts involving rotational motion, such as wheels or gears.
- **Formula**: Angular velocity (\( \omega \)) is calculated in radians per second and can be derived from revolutions per minute (rpm) by: \[\omega = \frac{2\pi \times \text{rpm}}{60}\]- **Units**: Typically measured in radians per second (\( \text{rad/s} \)).
In the exercise, the engine's rotational speed was given as \( 3750 \) rpm. Converting to radians per second gives approximately \( 392.7 ext{ rad/s} \). Understanding this conversion is crucial as it allows us to calculate power – that is, using angular velocity together with torque to assess engine performance.

Power is a measure of how quickly work can be done or energy can be transferred. In mechanical systems like engines, it's important to evaluate power to understand performance capabilities. This is where the concept of converting mechanical energy into power becomes vital.
- **Formula**: Power (\( P \)) can be found using:\[ P = \tau \times \omega \]This formula allows us to calculate power when we know the torque and angular velocity.
- **Units**: Power is often measured in Watts. In many applications, especially related to automotive engines, horsepower is used, where \( 1 ext{ hp} = 745.7 ext{ Watts} \).
In the given problem, by applying the known torque and angular velocity, the power calculated was \( 100165.5 \) Watts, which converts to approximately \( 134.3 \) horsepower. Power conversion like this is a vital check for understanding how effectively an engine can transform fuel into motion.