Problem 12 - Applied Physics

Question

Convert 285 rpm to \(\mathrm{rad} / \mathrm{s}\).

Step-by-Step Solution

Answer

285 rpm = approximately 29.85 rad/s.

Step 1: Understand the relationship

To convert revolutions per minute (rpm) to radians per second (\( \mathrm{rad/s} \)), we need to understand the relationship between these units. 1 revolution is equal to \( 2\pi \) radians, and there are 60 seconds in a minute.

Step 2: Apply the conversion factors

We want to convert 285 rpm to \( \mathrm{rad/s} \). First, convert revolutions to radians. Each revolution is \( 2\pi \) radians:\[ 285 \text{ rpm} \times 2\pi \text{ rad/rev} = 570\pi \text{ rad/min} \]

Step 3: Convert minutes to seconds

Next, convert the radians per minute to radians per second by dividing by 60, since there are 60 seconds in one minute:\[ 570\pi \text{ rad/min} \div 60 \text{ s/min} = \frac{570\pi}{60} \text{ rad/s} \]

Step 4: Simplify the expression

Simplify the expression from the previous step:\[ \frac{570\pi}{60} \text{ rad/s} = \frac{57\pi}{6} \text{ rad/s} \]Or approximately:\[ \frac{57\pi}{6} \approx 29.85 \text{ rad/s} \]

Key Concepts

Radians per Second

Radians per second (\( \mathrm{rad/s} \)) is a unit that measures angular velocity or how fast an object rotates. It describes how many radians an object moves through in one second. Here, radians as a unit pertain to the angle made by choosing the radius of the circle as the arc length. This naturally correlates to the circumference, where\( 2\pi \) radians make a complete revolution or a full circle.

If you picture a clock, imagine the second hand. As it moves from one tick mark to the next, it travels through an angle. If this angle is measured in radians and it happens in one second, it gives you the object's angular velocity in radians per second. Understanding this fundamental concept is crucial when converting between different units of angular speed.

Revolutions per Minute

Revolutions per minute (rpm) is another common unit to describe angular velocity, commonly used to specify speeds in various mechanical devices. When something spins one full revolution in a minute, its speed is 1 rpm. This concept is frequently applied in contexts like car engines and record players where spin speeds need to be precise.

Connecting rpm to radians per second involves knowing that each revolution is equivalent to\( 2\pi \) radians. Therefore, the transition from measuring revolutions in a minute to radians in a second requires careful calculation using this relationship. When converting, the equation revolves around changing minutes into seconds and revolutions into radians, making it a straightforward double conversion.

Conversion Factors

Conversion factors are crucial for moving between units. They represent the relationship between two types of measurements, allowing for a seamless transition. In the case of converting rpm to radically different \( \mathrm{rad/s} \), two conversion factors come into play: the number of radians in a revolution and the number of seconds in a minute.

• 1 revolution = \( 2\pi \) radians
• 1 minute = 60 seconds

To convert 285 rpm to radians per second, you'd first swap revolutions for radians and then adjust the time from minutes to seconds. Always ensure your conversion factors are aligned correctly so they cancel out other units appropriately, leaving the desired unit of measure.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point, usually the center of a circle. It's often expressed in radians per second (\( \mathrm{rad/s} \)) or revolutions per minute (rpm). This distinguishes it from linear velocity, which deals with straight-line movement.

Angular velocity provides insight into the rotational speed of an object, key in countless applications, including mechanics, physics, and engineering. When expressed in radians per second, you gain a direct correlation to the periodic and harmonic motion common in waves and rotations in science.

Understanding angular velocity is essential when interpreting how mechanisms, such as gears or spinning wheels, function. It provides a way to measure and predict motion outcomes in systems, helping to design and analyze anything from household appliances to complex industrial machineries.