Problem 21 - Applied Physics

Question

Find the angular velocity of a motor developing \(10 \overline{0} 0 \mathrm{~W}\) of power with a torque of \(15 \overline{0} \mathrm{~N} \mathrm{~m}\)

Step-by-Step Solution

Answer

The angular velocity is 66.67 radians per second.

Step 1: Understanding the Problem

To find the angular velocity of the motor, we know it develops a power of 10,000 Watts and has a torque of 150 Newton-meters. We can use the relationship between power, torque, and angular velocity given by the formula: \[ P = au \cdot \omega \] where \( P \) is the power, \( \tau \) is the torque, and \( \omega \) is the angular velocity (in radians per second).

Step 2: Rearranging the Formula

Since we need to find \( \omega \), rearrange the formula \( P = \tau \cdot \omega \) to solve for \( \omega \). This gives us: \[ \omega = \frac{P}{\tau} \]

Step 3: Substitute the Given Values

Now, substitute the given values for power and torque into the formula. We have \( P = 10,000 \) Watts and \( \tau = 150 \) Newton-meters, so: \[ \omega = \frac{10,000}{150} \]

Step 4: Calculate the Angular Velocity

Perform the division to find the angular velocity: \[ \omega = \frac{10,000}{150} = 66.67 \] This means the angular velocity \( \omega \) is 66.67 radians per second.

Key Concepts

Power

In physics, power is an essential concept that tells us how fast energy is used or transferred. It is measured in Watts (W), a unit named after James Watt, the inventor who made crucial improvements to the steam engine.
Power can be calculated using the formula: \[P = \tau \cdot \omega \]where \( P \) stands for power, \( \tau \) is torque, and \( \omega \) is angular velocity. In the context of rotational motion, like in a motor, power indicates how efficiently the motor converts energy into its rotating motion. Consider a motor that has a power rating of 10,000 Watts. This tells us the motor can efficiently use or transfer 10,000 Joules of energy each second to perform its work.

Torque

Torque is a measure of the twisting force that causes an object to rotate around an axis. It's an essential concept in understanding motor physics, as it's the "rotational equivalent" of linear force.
Torque is measured in Newton-meters (Nm) and is calculated by the formula:\[\tau = F \times r\]where \( F \) is the force applied, and \( r \) is the distance from the axis of rotation. In practical terms, a motor with higher torque can exert a stronger rotational force, which means it can do more work on heavier objects or achieve higher rotational speeds.
For example, in our exercise, the motor has a torque of 150 Nm, allowing it to apply significant rotational force.

Radians per Second

Radians per second is the standard unit of angular velocity. It measures how quickly something rotates or revolves around an axis. Angular velocity is crucial in motor physics as it describes the speed of the motor's rotation. To calculate angular velocity \( \omega \) from power and torque, use the formula:\[\omega = \frac{P}{\tau}\]In the given problem, where the power is 10,000 Watts and the torque is 150 Nm, the angular velocity calculates to 66.67 radians per second. This figure indicates the motor rotates approximately 66.67 radians every second, showcasing the speed of its operation.

Motor Physics

Motor physics involves understanding how motors work to convert electrical energy into mechanical rotation. This includes analyzing various factors such as power, torque, and angular velocity.
A motor operates efficiently when it can convert a high percentage of electrical energy into rotational motion with minimal energy loss.
Motors are often rated by their power capacity and torque to indicate their potential performance. - **Power Capacity**: Determines how much energy the motor can utilize. - **Torque**: Indicates the turning force the motor can exert. Understanding these ratings helps in selecting the appropriate motor for different machinery or tools based on their specific operational requirements.

Applied Physics Formulas

Applied physics formulas bridge theoretical concepts with practical applications. In motor physics, formulas like the power-torque-angular velocity relationship are used to design and optimize motor performance.This formula:\[P = \tau \cdot \omega\]allows engineers and students to predict how changes in one variable affect the others, making it easier to control motor functions.
For example, increasing torque while keeping power constant results in reduced angular velocity, altering the motor's speed but increasing its ability to handle heavier loads.