Question

Show that 1 rev \(/ \mathrm{min}=0.105 \mathrm{rad} / \mathrm{s}\).

Short answer

The conversion from revolutions per minute to radians per second is valid - 1 rev/min is indeed approximately equal to 0.105 rad/s.

Step-by-step solution

Understand the Units

A revolution (rev) is a full turn, equivalent to \(2\pi\) radians (rad). Furthermore, time units should be converted from minutes to seconds, since there are 60 seconds in a minute.

Calculate Radians per Second

Take the known value for a revolution (\(2\pi\) rad) and divide it by the number of seconds in a minute (60): \(rad/s = \frac{2\pi \, rad}{60 \, s}\)

Simplify the Radians per Second Expression

The expression simplifies to approximately 0.105 rad/s: \( \frac{2\pi \, rad}{60 \, s} \approx 0.105 \, rad/s \)

Key concepts

Radians Per Second

Radians per second is a unit of angular velocity, which is a measure of how quickly an object rotates or revolves around a central point. One radian per second means that the object will travel the distance of one radian on the circumference of the circle every second. To visualize a radian, imagine the radius of a circle being wrapped along its edge; the length covered is one radian.

Since a complete revolution of a circle contains exactly \(2\textpi\) radians, understanding that the unit 'radian per second' inherits this circular nature is crucial in establishing the angular speed. For example, if an object completes half a revolution in one second, its angular speed would be \(\textpi\) radians per second because it has traveled \(\textpi\) radians in that time.

Revolution Per Minute

A revolution per minute (rpm) is another common unit used to express angular velocity, particularly in mechanical and engineering disciplines. It specifies how many complete turns an object makes in one minute.

The use of rpm is widespread in various applications such as measuring the speed of a motor or the rotation of a wheel. One revolution equates to a 360-degree turn which is mathematically equivalent to \(2\textpi\) radians. The transition from revolutions per minute to radians per second requires conversion because the two units incorporate different time frames—minutes versus seconds. Understanding this differentiation permits the accurate assessment of rotational speeds in various scientific and practical scenarios.

Unit Conversion

Unit conversion is a fundamental concept in physics and engineering that allows for the comparison and computation of quantities in different units of measurement. Converting from revolutions per minute to radians per second involves a two-step process: first converting the time unit from minutes to seconds, and then converting the angular measurement from revolutions to radians.

The conversion factor from minutes to seconds is 60, as there are 60 seconds in a minute. Moreover, since one revolution is \(2\textpi\) radians, this value is used to convert between the two angular units. To ensure accuracy and coherence when comparing or calculating velocities, mastering unit conversion practices is essential; doing so facilitates communication and collaboration across various fields that rely on precise measurement values.