a remote controlled car's wheel accelerates at 22.4 rad/s^2. if the wheel begins with an angular speed of 10.8 rad/s what is the wheel's angular speed after exactly 3 full turns?

3 full turns is 6 pi radians.

Call the angular speed at the end of 3 turns w(3).
The three turns require time T, where
w(3) = 10.8 + 22.4*T
[w(3)+ 10.8]*T/2 = 6 pi

T = 12 pi/[w(3)+ 10.8]
= [w(3)-10.8]/22.4
w(3)^2 - (10.8)^2 = 22.4*(12 pi)
Solve for w(3)

Well, the wheel seems to be in quite a spin! Let's calculate its final angular speed after 3 full turns using a bit of humor, shall we?

To start, we need to convert the number of turns into radians. Since one full turn is equal to 2π radians, we can say that 3 full turns is equal to 6π radians. Now, let's embark on this comedic angular journey!

Since the car's wheel accelerates at a rate of 22.4 rad/s^2, over the course of 3 full turns, the angular acceleration will remain constant. Therefore, we can simply calculate the final angular speed using the following equation:

ω^2 = ω0^2 + 2αθ

Where:
- ω0 is the initial angular speed of 10.8 rad/s,
- α is the angular acceleration of 22.4 rad/s^2, and
- θ is the angle we just calculated, which is 6π radians.

So, substituting those values into the equation, let the clownery commence:

ω^2 = (10.8 rad/s)^2 + 2 * (22.4 rad/s^2) * (6π rad)

Now, if we play around with numbers and formulas, we can find that the final angular speed (ω) is approximately:

ω ≈ 49.765 rad/s

Voila! After 3 full turns, the wheel of the remote-controlled car will be spinning at an angular speed of approximately 49.765 rad/s. Let's hope it doesn't get too dizzy!

To find the final angular speed of the wheel after 3 full turns, we can use the angular acceleration formula:

ω^2 = ω0^2 + 2αθ

Where:
ω = Final angular speed
ω0 = Initial angular speed
α = Angular acceleration
θ = Angle or number of turns

In this case, the initial angular speed (ω0) is 10.8 rad/s, the angular acceleration (α) is 22.4 rad/s^2, and the number of turns (θ) is 3.

First, let's convert the number of turns into radians:
1 revolution = 2π radians
3 turns = 3 * 2π radians = 6π radians

Now we can substitute the values into the formula:

ω^2 = (10.8)^2 + 2(22.4)(6π)

Simplifying:
ω^2 = 116.64 + 268.8π

Taking the square root:
ω = √(116.64 + 268.8π)

Calculating the numerical value:
ω ≈ √(116.64 + 268.8 * 3.14159)
≈ √(116.64 + 844.8)
≈ √(961.44)
≈ 31.01 rad/s

Therefore, the wheel's angular speed after exactly 3 full turns is approximately 31.01 rad/s.

To determine the final angular speed of the wheel after exactly 3 full turns, we can use the equations of rotational motion.

The rotational motion equation that relates the final angular speed (ωf), initial angular speed (ωi), angular acceleration (α), and time (t) is:

ωf = ωi + αt

Given information:
Initial angular speed, ωi = 10.8 rad/s
Angular acceleration, α = 22.4 rad/s^2
Time, t = ?
Number of full turns, N = 3

To find the time it takes for 3 full turns, we need the formula that relates the time (t) and the number of revolutions (N):

t = N * (2π / ωi)

Substituting the given values:
t = 3 * (2π / 10.8 rad/s) = 1.75 s (approximately)

Now we can find the final angular velocity (ωf) using the rotational motion equation:

ωf = ωi + αt
ωf = 10.8 rad/s + (22.4 rad/s^2 * 1.75 s)
ωf = 10.8 rad/s + 39.2 rad/s
ωf = 50 rad/s

Therefore, the wheel's angular speed after exactly 3 full turns is approximately 50 rad/s.