A merry-go-round accelerating uniformly from rest achieves its operating speed of 3.4 rpm in 2 revolutions. What is the magnitude of the angular acceleration?

4 pi radians = distance gone

3.4 *2 pi/60 = final speed in radians/s
average speed = half of final = 3.4 pi/60
so
4 pi = (3.4 pi/60)t
so
t = 4*60/3.4

v = Vi + a t
3.4*2 pi/60 = a (4*60/3.4)
a = (3.4^2 /6^2)(pi/2)

The magnitud of angular 6.8

To find the magnitude of the angular acceleration, we need to use the formula:

ωf = ωi + αt

Where:
ωf = final angular velocity
ωi = initial angular velocity
α = angular acceleration
t = time

Given that the merry-go-round accelerates uniformly from rest, the initial angular velocity (ωi) is 0.

We are given the final angular velocity (ωf) as 3.4 rpm. To convert this to radians per second, we use the conversion factor:

1 revolution = 2π radians

So, 3.4 rpm is equivalent to:

3.4 rpm * (2π radians / 1 revolution) = 3.4 * 2π radians/minute

Next, we need to convert this to radians per second by dividing by 60 since there are 60 seconds in a minute:

(3.4 * 2π radians/minute) / 60 seconds = 3.4 * 2π / 60 radians/second

Now we can substitute the values into the formula:

3.4 * 2π / 60 = 0 + α * 2

Simplifying the equation, we have:

(3.4 * 2π) / 60 = 2α

Dividing both sides by 2, we get:

(3.4 * 2π) / (60 * 2) = α

Simplifying further, we have:

(3.4 * π) / 60 = α

Calculating this expression gives us:

α ≈ 0.056 rads/s²

Therefore, the magnitude of the angular acceleration is approximately 0.056 radians per second squared.

To find the magnitude of the angular acceleration, we need to use the formula:

Angular acceleration (α) = (Final angular velocity (ω) - Initial angular velocity (ω₀)) / Time taken (t)

In this case, the final angular velocity (ω) is given as 3.4 rpm. However, we need to convert this to radians per second since angular acceleration is typically measured in radians per second squared. To convert from rpm to radians per second, we use the conversion factor of 2π rad per minute.

First, let's convert 3.4 rpm to radians per second:

3.4 rpm * (2π rad / 1 min) * (1 min / 60 s) = 3.4 * 2π / 60 rad/s ≈ 0.355 rad/s

Now, let's find the initial angular velocity (ω₀) which is the angular velocity at the start when the merry-go-round is at rest. Since it starts from rest, ω₀ = 0 rad/s.

The time taken (t) to achieve the operating speed of 3.4 rpm in 2 revolutions is given as 2 revolutions. We need to convert this to seconds by considering that one revolution is the time taken for the merry-go-round to complete a full circle, or 360 degrees.

Time taken (t) = 2 revolutions * (1 revolution / 360 degrees) * (1 degree / 3600 seconds 1 second / 60 minutes) ≈ 2 * 2π / 360 rad/s

Now, we plug in the values into the formula for angular acceleration:

α = (ω - ω₀) / t
= (0.355 rad/s - 0 rad/s) / (2 * 2π / 360 rad/s)
= (0.355 rad/s) / (2 * 2π / 360 rad/s)
≈ 0.355 rad/s / (6.28 rad/s)
≈ 0.057 rad/s²

Therefore, the magnitude of the angular acceleration is approximately 0.057 rad/s².