Problem 11 - Applied Physics

Question

An engine has torque of \(550 \mathrm{~N} \mathrm{~m}\) at \(8.3 \mathrm{rad} / \mathrm{s}\). What power in watts does it develop?

Step-by-Step Solution

Answer

The engine develops 4565 watts.

Step 1: Understand the Relationship Between Torque, Angular Velocity, and Power

To find the power output of an engine using torque and angular velocity, you need to use the formula for power in rotational systems: \[ P = \tau \times \omega \] where \( P \) is power in watts, \( \tau \) is torque in newton-meters (Nm), and \( \omega \) is angular velocity in radians per second (rad/s).

Step 2: Substitute Known Values into the Power Equation

Given that the torque \( \tau = 550 \) Nm and the angular velocity \( \omega = 8.3 \) rad/s, substitute these values into the power formula:\[ P = 550 \times 8.3 \]

Step 3: Calculate the Power

Multiply the torque by the angular velocity to find the power:\[ P = 550 \times 8.3 = 4565 \] Thus, the power developed by the engine is 4565 watts.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object. Imagine trying to open a door. The further away from the hinge you push, the easier it is to open it. This is because you apply torque when you exert force at a distance from the point of rotation. The formula for torque is \( \tau = r \times F \times \sin(\theta) \), where \( r \) is the distance from the pivot point, \( F \) is the force applied, and \( \theta \) is the angle between the force and the lever arm. In simple systems like engines, torque is often applied parallel to the axis of rotation, making the sine function unnecessary, simplifying to \( \tau = r \times F \). Torque is measured in newton-meters (Nm) and plays a critical role in determining how much rotational force an engine can produce.

Angular Velocity

Angular velocity describes how fast an object rotates or spins. It's like the speedometer of a rotating system, indicating how many radians it moves through per unit of time. Think of the second hand on a clock: as it moves from one second to the next, it travels through a certain angle measured in radians. The formula for angular velocity is \( \omega = \frac{\Delta \theta}{\Delta t} \), where \( \Delta \theta \) is the change in angular position in radians, and \( \Delta t \) is the time it took for that change. Common units for angular velocity are radians per second (rad/s). In many practical applications, such as engines, knowing the angular velocity is essential for calculating other important quantities like kinetic energy and power.

Power Calculation

Power in physics refers to how quickly work is done or energy is transferred. In rotational systems like engines, where both torque and angular velocity are involved, power calculation is essential to determine the engine's output. The formula used is \( P = \tau \times \omega \), where \( P \) is the power in watts, \( \tau \) is the torque in newton-meters, and \( \omega \) is the angular velocity in radians per second. This formula shows us that power is a product of both the force producing the rotation and the speed of the rotation. Substituting the given numbers, as in the original exercise, directly into this formula allows us to calculate the engine's power output easily.

Rotational Systems

Rotational systems comprise parts or mechanisms that rotate around a central axis. These systems are essential in various applications, from car engines to wind turbines. Understanding the dynamics of rotational systems revolves around several key concepts, such as torque, angular velocity, and inertia. Unlike linear systems, where mass and velocity define the movement, rotational systems use moments of inertia and angular quantities. This means equations and concepts can slightly differ from their linear counterparts. Overall, studying rotational systems provides insights into improving functionality and efficiency in machinery that feature spinning or rotating components.