What is the angular velocity of the wheel if it rotates at 120 rev/min?
Understand the Problem
The question is asking us to convert the rotational speed of a wheel from revolutions per minute (rev/min) to angular velocity in radians per second (rad/s). To solve this, we'll need to use the conversion factor that 1 revolution equals $2 heta ext{pi}$ radians and convert minutes to seconds.
Answer
The angular velocity in radians per second is given by $ \frac{N \pi}{30} $.
Answer for screen readers
The angular velocity in radians per second is given by:
$$ \text{Angular velocity (rad/s)} = \frac{N \pi}{30} $$
Steps to Solve
-
Identify the conversion factors
To convert revolutions to radians, we use the fact that 1 revolution is equal to $2\pi$ radians. To convert minutes to seconds, we note that 1 minute equals 60 seconds. -
Write down the initial value
Let’s say the initial rotational speed of the wheel is given as $N$ rev/min. We will use this to perform our conversions. -
Convert revolutions to radians
To convert the revolutions per minute to radians per minute, we can use the conversion: $$ \text{Radians per minute} = N \text{ rev/min} \times 2\pi \text{ rad/rev} $$ -
Convert minutes to seconds
Now, we can convert the radians per minute to radians per second by dividing by 60 (because there are 60 seconds in a minute): $$ \text{Angular velocity (rad/s)} = \frac{\text{Radians per minute}}{60} $$ -
Combine the conversions
Putting the two conversions together, we have: $$ \text{Angular velocity (rad/s)} = \frac{N \times 2\pi}{60} $$ -
Simplify
The final expression can be simplified: $$ \text{Angular velocity (rad/s)} = \frac{N \pi}{30} $$
The angular velocity in radians per second is given by:
$$ \text{Angular velocity (rad/s)} = \frac{N \pi}{30} $$
More Information
This formula allows you to convert any given rotational speed from revolutions per minute to angular velocity in radians per second. Remember, this conversion is particularly useful in physics and engineering to analyze rotating objects.
Tips
- Confusing the conversion from revolutions to radians: Always remember that 1 revolution equals $2\pi$ radians.
- Forgetting to convert minutes to seconds correctly may lead to errors in the final answer. Always divide by 60 when converting.
