Question

Angular velocity describes the rate of change in a central angle over time. For example, the change could be expressed in revolutions per minute (rpm), radians per second, degrees per hour, and so on. All that is required is an angle measurement expressed over a unit of time. a) Earth makes one revolution every \(24 \mathrm{h}\). Express the angular velocity of Earth in three other ways. b) An electric motor rotates at 1000 rpm. What is this angular velocity expressed in radians per second? c) A bicycle wheel completes 10 revolutions every 4 s. Express this angular velocity in degrees per minute.

Short answer

Earth's angular velocity is 0.0417 rev/h, \(7.2722 \times 10^{-5}\) rad/s, 15 deg/h. Motor's angular velocity is 104.72 rad/s. Bicycle's angular velocity is 54000 deg/min.

Step-by-step solution

Understanding Angular Velocity

Angular velocity is the rate of change of an angle over time. It is often measured in units such as revolutions per minute (rpm), radians per second (rad/s), and degrees per hour (deg/h).

Calculate Earth's Angular Velocity in Revolutions per Hour (rev/h)

Earth makes one full revolution every 24 hours. Therefore, Earth's angular velocity in revolutions per hour is: \[ \text{Angular velocity} = \frac{1 \text{ revolution}}{24 \text{ hours}} = 0.0417 \text{ rev/h} \]

Calculate Earth's Angular Velocity in Radians per Second (rad/s)

We know that one revolution is equal to \(2\pi\) radians. Thus: \[ \text{Angular velocity} = \frac{2\pi \text{ radians}}{24 \times 3600 \text{ seconds}} \approx 7.2722 \times 10^{-5} \text{ rad/s} \]

Calculate Earth's Angular Velocity in Degrees per Hour (deg/h)

One full revolution is 360 degrees. Therefore: \[ \text{Angular velocity} = \frac{360 \text{ degrees}}{24 \text{ hours}} = 15 \text{ deg/h} \]

Convert Electric Motor RPM to Radians per Second

Given an electric motor rotating at 1000 rpm, we convert this to radians per second: \[ \text{Angular velocity} = 1000 \times \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{2000\pi}{60} \approx 104.72 \text{ rad/s} \]

Convert Bicycle Wheel Revolutions to Degrees per Minute

Given that a bicycle wheel completes 10 revolutions every 4 seconds, first find the number of revolutions per minute: \[ \text{Revolutions per minute} = \frac{10 \text{ revolutions}}{4 \text{ seconds}} \times 60 = 150 \text{ rev/min} \] Then, convert to degrees per minute: \[ 150 \times 360 \approx 54000 \text{ deg/min} \]

Key concepts

Revolutions per Minute (RPM)

Revolutions per minute (RPM) is a unit to measure angular velocity. It tells us how many complete rotations (revolutions) an object makes in one minute. It's a common unit when dealing with engines, motors, and other rotating machinery.
Imagine a bike wheel that turns 5 times in a minute. It has an angular velocity of 5 RPM.
To calculate RPM, simply count the number of full rotations in one minute.

• Example: An electric motor running at 1000 RPM completes 1000 full rotations in one minute.
• Conversion to other units: You can convert RPM to other angular velocity units like radians per second or degrees per second.

In maths, we often need to convert RPM to radians per second. One full revolution equals \(2\pi\) radians. So, the formula to convert RPM to radians per second is:
\( \text{rad/s} = \frac{2\pi \times \text{RPM}}{60} \)
This formula helps us express RPM in a different yet meaningful unit for various calculations.

Radians per Second (Rad/s)

Radians per second (Rad/s) is another way to measure angular velocity. Instead of rotations, it uses radians, a unit derived from the radius of a circle. One full revolution is equivalent to \(2\pi\) radians.
This unit is commonly used in physics and engineering. It provides a precise measure, especially for small and high-speed rotations.

• Example: If a fan blade rotates at 6.28 rad/s, it makes one full revolution per second, since \(2\pi \approx 6.28 \) radians.
• Applications: It's useful in calculating circular motion and oscillations.

To convert radians per second to RPM, we reverse the earlier formula:
\( \text{RPM} = \frac{60 \times \text{rad/s}}{2\pi} \)
This conversion is useful when analyzing data from different systems or units.

Degrees per Hour (Deg/h)

Degrees per hour (Deg/h) measures how many degrees an object rotates in one hour. This unit is helpful for slower rotations, like the Earth's rotation.
We know one full circle is 360 degrees. So, if an object makes one full rotation in 24 hours, like the Earth, its angular velocity is 15 degrees per hour.

• Example: Earth rotates at 15 degrees per hour, completing a full 360-degree rotation in a 24-hour day.
• Usage: Ideal for astronomy, clock mechanisms, and slower rotational systems.

To convert degrees per hour to radians per second, we use the following steps:
1. Convert degrees to radians (\text{1 degree} = \frac{\pi}{180} radians\br>2. Convert hours to seconds (\text{1 hour} = 3600 seconds)
Then:
\( \text{rad/s} = \frac{\pi \times Deg/h}{180 \times 3600} \)
This method accurately translates slower rotational measurements to a standard unit for more precise calculations.