A centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. If a point located 8.00 cm from the axis of rotation of the centrifuge moves with a speed of 150 m/s when the centrifuge is at full speed, how many revolutions does the centrifuge make as it goes from rest to its final angular speed?

To solve this problem, we need to use the equations of motion for rotational motion.

First, let's find the final angular speed of the centrifuge. We are given that it takes 100 s to reach this speed.

Using the equation:

angular acceleration (α) = (final angular speed (ω) - initial angular speed (ω₀)) / time (t)

Since the centrifuge starts at rest, the initial angular speed (ω₀) is 0. Rearranging the equation, we get:

ω = α * t

Given the time (t) as 100 s, we can't determine the angular acceleration (α) from the given information. Hence, we need more information to calculate the number of revolutions the centrifuge makes.

However, if we assume that the angular acceleration is constant, we can proceed with the calculations using the given information.

Let's find the angular speed (ω) when the point located 8.00 cm from the axis of rotation moves with a speed of 150 m/s.

Using the formula:

v = ω * r

Where v is the linear speed, ω is the angular speed, and r is the distance from the axis of rotation.

Converting the 8.00 cm to meters (0.08 m) and substituting the values, we have:

150 m/s = ω * 0.08 m

Solving for ω, we get:

ω = 150 m/s / 0.08 m = 1875 rad/s

Now that we have the angular speed (ω) at the full speed, we can calculate the number of revolutions the centrifuge makes.

One revolution is equal to 2π radians. So the number of revolutions is given by:

Number of revolutions = ω / (2π)

Substituting the value of ω, we have:

Number of revolutions = 1875 rad/s / (2π) = 297.7 revolutions (approximately)

Therefore, the centrifuge makes approximately 297.7 revolutions as it goes from rest to its final angular speed, assuming a constant angular acceleration.

To find the number of revolutions the centrifuge makes as it goes from rest to its final angular speed, we need to use the concepts of angular acceleration, angular speed, and radius.

Let's start by identifying the relevant information given in the problem:

- Time taken to spin up: t = 100 s
- Radius of rotation: r = 8.00 cm = 0.08 m
- Speed of the point at full speed: v = 150 m/s

First, we need to calculate the angular acceleration (α) of the centrifuge. We can use the formula:

α = (ω - ω0) / t

Where:
- α is the angular acceleration
- ω0 is the initial angular speed (0 because the centrifuge is at rest in the beginning)
- ω is the final angular speed

Next, we can calculate the final angular speed using the formula:

ω = v / r

Where:
- ω is the angular speed
- v is the linear speed
- r is the radius

Now, let's substitute the given values into the formulas:

α = (150 m/s - 0 m/s) / 100 s
ω = (150 m/s) / (0.08 m)

Simplifying these equations gives us:

α = 1.5 rad/s^2
ω = 1875 rad/s

Finally, we can find the number of revolutions by using the formula:

Number of revolutions = (ω^2) / (2πα)

Plugging in the values we have:

Number of revolutions = (1875 rad/s)^2 / (2π * 1.5 rad/s^2)

Evaluating the above expression gives us the final answer.

v=rω

=>
150 m/s = 0.08m*ω
θ=150/0.08=1875 rad/s
Average ω=1875/2=937.5 rad/s
Number of revolutions
=937.5 rad/s *100s /(2π rad/rev)
=93750/2π
=14921 rev. approx.

Check my arithmetic.