A wheel turns through 90.0 revolutions in 15.0 sec. The angular speed at the end of this 15.0 sec period is 10.0 rev/s. Assuming that the angular acceleration is constant, calculate a) the angular speed at the beginning of the 15.0 sec interval, and b) the amount of time that elapsed from when the wheel was at rest and the beginning of the 15.0 sec interval.

avg angular speed=90rev/15sec=6 rev/sec

but avg ang speed=(wf+wi)/2
or 6=(10+wi)/2 so wi= 2 rpm

b. If it took 15 sec to increase by 8 rpm, it took 15*2/8= sec to get to 2rpm.

a) Well, to find the angular speed at the beginning of the 15.0 sec interval, we can use the formula for angular acceleration:

ωf = ωi + αt

Where:
ωf is the final angular speed (10.0 rev/s),
ωi is the initial angular speed (what we're trying to find),
α is the angular acceleration (which is constant),
and t is the time interval (15.0 sec).

Now, we know that the wheel turns through 90.0 revolutions in 15.0 sec, so we can find the angular acceleration by dividing the change in angular speed by the time interval:

α = Δω / t = (10.0 rev/s - 0 rev/s) / (15.0 sec) = 10.0/15.0 rev/(s^2)

Substituting all these values into the first equation, we have:

10.0 rev/s = ωi + (10.0/15.0 rev/(s^2))(15.0 sec)

Simplifying the equation:

10.0 rev/s = ωi + 10.0 rev/s

Subtracting 10.0 rev/s from both sides:

0 rev/s = ωi

So, the angular speed at the beginning of the 15.0 sec interval is 0 rev/s.

b) Now, to find the amount of time that elapsed from when the wheel was at rest to the beginning of the 15.0 sec interval, we have to use another formula:

θ = ωi*t + (1/2)αt^2

Where:
θ is the angle turned (90.0 revolutions),
ωi is the initial angular speed (0 rev/s),
α is the angular acceleration (10.0/15.0 rev/(s^2)),
and t is the time interval.

Let's solve for t by plugging in the given values:

90.0 rev = (0 rev/s)t + (1/2)(10/15 rev/(s^2))(t^2)

Simplifying the equation:

90.0 rev = (5/15)(t^2)

Dividing both sides by 5/15:

t^2 = 270.0 rev / (5/15 rev/(s^2)) = 18.0 s^2

Taking the square root of both sides:

t = √(18.0 s^2)

Calculating the square root:

t ≈ 4.24264 s

Therefore, the amount of time that elapsed from when the wheel was at rest to the beginning of the 15.0 sec interval is approximately 4.24264 seconds.

Hope that helps, but if it didn't, don't worry - I'm just a clown bot trying to make you smile!

To solve this problem, we can use the equations of rotational motion. Let's denote the angular speed at the beginning of the 15.0 sec interval as ω₁, and the time elapsed from when the wheel was at rest to the beginning of the 15.0 sec interval as t.

a) To calculate the angular speed at the beginning of the 15.0 sec interval (ω₁), we can use the equation:

ω = ω₀ + αt

where ω₀ is the initial angular speed (unknown in this case), α is the angular acceleration (unknown in this case), and t is the time interval.

We are given that the wheel turns through 90.0 revolutions in 15.0 sec, so we can convert this to radians:

90.0 revolutions * 2π radians/revolution = 90.0 * 2π radians = 180π radians

We also know that the angular speed at the end of the 15.0 sec interval is 10.0 rev/s, which we can convert to radians:

10.0 rev/s * 2π radians/revolution = 20π radians/s

Using these values:

20π radians/s = ω₁ + α(15.0 sec)

Since we are assuming that the angular acceleration is constant, α can be calculated using the equation:

α = (ω - ω₀) / t

We know ω = 20π radians/s, ω₀ is unknown, and t = 15.0 sec. Plugging in these values:

α = (20π radians/s - ω₀) / 15.0 sec

Now we can substitute this value of α into the first equation:

20π radians/s = ω₁ + [(20π radians/s - ω₀) / 15.0 sec] * 15.0 sec

Simplifying:

20π radians/s = ω₁ + 20π radians/s - ω₀

Subtracting 20π radians/s from both sides:

0 = ω₁ - ω₀

So, the angular speed at the beginning of the 15.0 sec interval (ω₁) is ω₀.

b) Now, to calculate the amount of time that elapsed from when the wheel was at rest to the beginning of the 15.0 sec interval (t), we can rearrange the equation:

α = (ω - ω₀) / t

as:

t = (ω - ω₀) / α

Using the values ω = 20π radians/s and ω₀ = 0 (since the wheel starts from rest):

t = (20π radians/s - 0) / [(20π radians/s - 0) / 15.0 sec]

Simplifying:

t = 15.0 sec

Hence, the amount of time that elapsed from when the wheel was at rest to the beginning of the 15.0 sec interval is 15.0 sec.

To find the angular speed at the beginning of the 15.0 sec interval, we can use the formula for angular acceleration:

angular acceleration = (angular speed at the end - angular speed at the beginning) / time taken

Given:
angular acceleration = constant
angular speed at the end = 10.0 rev/s
time taken = 15.0 sec

Substituting these values into the formula, we have:

angular acceleration = (10.0 rev/s - angular speed at the beginning) / 15.0 sec

Now, let's solve for the angular speed at the beginning:

angular acceleration * time taken = 10.0 rev/s - angular speed at the beginning

angular speed at the beginning = 10.0 rev/s - (angular acceleration * time taken)

To find the amount of time that elapsed from when the wheel was at rest and the beginning of the 15.0 sec interval, we need to calculate the time it took for the wheel to make 90.0 revolutions.

Given:
Number of revolutions = 90.0 rev
Angular speed = 10.0 rev/s
Time taken = ?

We can use the formula:

Number of revolutions = Angular speed * Time taken

Rearranging the formula to solve for Time taken:

Time taken = Number of revolutions / Angular speed

Substituting the given values into the formula, we get:

Time taken = 90.0 rev / 10.0 rev/s

Time taken = 9.0 s

Therefore, the angular speed at the beginning of the 15.0 sec interval is:

angular speed at the beginning = 10.0 rev/s - (angular acceleration * time taken)
= 10.0 rev/s - (angular acceleration * 15.0 s)

And the amount of time that elapsed from when the wheel was at rest and the beginning of the 15.0 sec interval is:

time elapsed = Time taken - time for 15.0 sec interval
= 9.0 s - 15.0 s
= -6.0 s

It's important to note that the negative sign in the time elapsed indicates that the 15.0 sec interval is before the wheel came to rest.