Question

Dragonflies often dip into the water on the surface of a lake or pond, presumably to drink or to clean themselves. Recently, it has been discovered that after dipping the dragonflies fly straight upward, then pitch forward and tumble head over tail several times until the water has been shed from their bodies. Observations show that the dragonflies complete one revolution every $0.017\mathrm{s}$ during this "spin dry" maneuver. What is their angular speed in radians per second and revolutions per minute? (For comparison, a typical car engine runs at about $1500\mathrm{rpm}$.)

Short answer

The angular speed is about 369.60 rad/s or 3533 rpm.

Step-by-step solution

Understanding Angular Speed

Angular speed ($\omega$) is the rate at which an object rotates or revolves. It's measured in radians per second (rad/s) and sometimes converted to revolutions per minute (rpm). First, we need to find the angular speed in radians per second, and then convert it to revolutions per minute.

Identify Given Values

We are given that a dragonfly completes one revolution every $0.017\text{s}$. A full revolution is $2\pi \text{ radians}$. We will use these values to calculate the angular speed.

Calculate Angular Speed in Radians per Second

The formula for angular speed in radians per second is $\omega =\frac{\theta }{t}$, where $\theta$ is the angle in radians and $t$ is the time. Substituting $\theta =2\pi$ and $t=0.017$, we have:$$\omega =\frac{2\pi }{0.017}\text{rad/s}$$Calculate $\omega$ to find the angular speed in radians per second.

Calculate the Result

Using a calculator, we find:$$\omega =\frac{6.2832}{0.017}\approx 369.60\text{ rad/s}$$Thus, the angular speed is approximately $369.60\text{ rad/s}$.

Convert Radians per Second to Revolutions per Minute

To convert from rad/s to rpm, recall that there are $2\pi$ radians in one revolution and $60$ seconds in one minute. Use the conversion formula:$$\text{rpm}=\frac{\omega }{2\pi }\times 60$$Plug the value of $\omega$ (369.60 rad/s) into this formula to find rpm.

Calculate the Conversion

Substitute and calculate as follows:$$\text{rpm}=\frac{369.60}{2\pi }\times 60$$Using a calculator, this results in approximately $3533\text{ rpm}$. Thus, the angular speed is approximately $3533\text{ rpm}$.

Key concepts

Radians per Second

Radians per second is a unit of angular speed that tells us how fast something is rotating or spinning. Imagine a circle where one full rotation equals to 360 degrees or in terms of radians, it equals to $2\pi$ radians.
For a rotating object, if it completes one full circle in one second, its angular speed is $2\pi$ radians per second. This means in one second, the object has travelled an angular distance equal to the circumference of the circle (or the full angle in radians).

• This unit is widely used in physics because of its simplicity in calculations, especially when dealing with sinusoidal functions, which naturally arise with circular motion.
• Radians are preferred because they create seamless integration with other SI units such as meters and seconds without requiring additional conversion factors.

Understanding angular speed in radians per second forms a key component in describing rotational motion comprehensively.

Revolutions per Minute

Revolutions per minute (rpm) is another popular unit for measuring angular speed, especially in scenarios involving machinery or engines. It tells us how many full rotations an object completes in one minute.
To better understand, think of a wheel completing a full turn: one revolution. Now imagine how many times it can do this within a minute—that number is its speed in rpm.
This unit is particularly helpful for comparing mechanical speeds — for example, comparing a dragonfly's "spin dry" maneuver to the functioning of a car's engine running at approximately 1500 rpm.

• Rpm is easier to relate to more intuitive physical scenarios like hands of a clock or cars’ rotations.
• It provides an accessible way to communicate rotational speeds across diverse applications in both engineering and day-to-day contexts.

Conversion Formula

Sometimes it is helpful to convert angular speed from one unit to another, such as from radians per second (rad/s) to revolutions per minute (rpm). For this, a specific conversion formula is used.
This conversion involves two main numbers: $2\pi$ radians per revolution and $60$ seconds per minute.
The formula to perform this conversion is:$$\text{rpm}=\frac{\omega }{2\pi }\times 60$$where $\omega$ is the angular speed in rad/s.

• First, dividing $\omega$ by $2\pi$ gives the number of complete revolutions per second.
• Then by multiplying by $60$, you convert that number into revolutions per minute.

This process assures that both radian and time conversions are accounted for, providing a seamless transition between these two units in any calculation.

Physics Problem Solving

When tackling physics problems involving angular speed, breaking them down into smaller parts is beneficial for a better understanding. In this problem, we first need to identify what information has been given and what is required.
The known values, such as the time it takes for one full revolution and the definition of a full revolution in radians, help us set up our initial calculations in rad/s.
Once the angular speed is determined in rad/s, we can convert it to rpm, if needed, using the conversion formula.
Here are a few tips for similar problems:

• Always ensure clarity on units being used. Units like radians, seconds, and minutes set the stage for correct calculations.
• Laying out each step methodically helps in validating the final answer, ensuring it aligns with both the physical intuition and numerical result.
• Incorporating checks like conversion back to original units can help verify the accuracy of the calculations.

Through such structured problem solving, not only are solutions derived but also key physics concepts are reinforced, aiding deeper comprehension.