Some dragonflies splash down onto the surface of a lake to clean themselves. After this dunking, the dragonflies gain altitude, and then spin rapidly at about 1100 rpm to spray the water off their bodies. When the dragonflies do this "spin-dry," they tuck themselves into a "ball" with a moment of inertia of 2.1×10−7kg⋅m2.
1/2(2.1*10^-7)(115.1917306)^2 =0.00139 kgm^2/s^2
kgm^2/s^2
is also J for units
1/2 (2.1*10^-7)*[(2pi*1100/60)^2]
Dont forget to square the answer when you multiply 2pi with the conversion of rpm because it is essentially k=1/2 Iw^2
And use J for units
Well, well, well, look at you getting all scientific with your calculations! I don't know what's more impressive - the fact that you're doing math or the fact that you're doing math about dragonflies. Either way, I'm impressed!
Now, let me put on my thinking, uh, wig, and see if I can attempt to decipher what you just said. It seems like you're calculating the kinetic energy of the spinning dragonflies, which is pretty cool. With a moment of inertia of 2.1×10−7 kg⋅m2, and spinning at 1100 rpm (which is about 115.1917306 radians per second, in case you care), you've determined that the kinetic energy is 0.00139 kgm^2/s^2. Fascinating!
It's amazing to think about these little dragonflies going for a spin to dry themselves off. Who needs towels when you have a built-in spin cycle, right? I bet they're the envy of all the other insects in the animal kingdom, wishing they had long tails to spin around like whirling dervishes. Keep up the good work with your calculating, you math wizard, you!
To explain how this answer was obtained, let's break it down step by step.
First, let's consider the "spin-dry" motion of the dragonflies. The moment of inertia (I) of an object represents its resistance to changes in its rotational motion. It depends on both the mass of the object and its spatial distribution of mass. In this case, the dragonflies tuck themselves into a "ball" to spin-dry, so we assume that their mass is distributed uniformly and can be considered as a solid sphere.
The moment of inertia of a solid sphere is given by the expression I = (2/5) * m * r^2, where m is the mass of the sphere and r is its radius.
In the given question, the moment of inertia of the dragonflies' ball is 2.1 × 10^(-7) kg⋅m^2. We can rearrange the formula and solve for the radius (r):
I = (2/5) * m * r^2
r^2 = (5/2) * (I / m)
r = sqrt((5/2) * (I / m))
Now, we need to calculate the radius of the dragonflies' ball.
Since there is no information about the mass (m) of the dragonflies, we cannot calculate the radius with the given information alone.
However, in the next part of your question, you seem to be calculating the kinetic energy (KE) of the spinning dragonflies, assuming that their mass is 0.00139 kg and their moment of inertia is 2.1 × 10^(-7) kg⋅m^2.
The kinetic energy of a rotating object is given by the formula KE = (1/2) * I * ω^2, where ω is the angular velocity in radians per second.
In the given calculation, you used the formula with the calculated moment of inertia (I) and the angular velocity (ω) of 1100 rpm. However, it seems like the conversion from rpm to radians per second is missing in your calculation.
To convert the angular velocity from rpm to radians per second, we need to multiply it by (2π/60), since there are 2π radians in a full revolution and there are 60 seconds in a minute.
Let's redo the calculation with the correct conversion:
ω = 1100 rpm * (2π/60) rad/s = 115.1917306 rad/s
Now we can use the formula for kinetic energy:
KE = (1/2) * I * ω^2 = (1/2) * (2.1 × 10^(-7) kg⋅m^2) * (115.1917306 rad/s)^2
≈ 0.00139 kg⋅m^2/s^2
Therefore, the calculated value of 0.00139 kg⋅m^2/s^2 seems to be correct, assuming the given mass and moment of inertia values. However, it is important to note that the calculation uses an assumed mass value that is not explicitly given in the question.