(b)A centrifuge has a maximum rotation rate of 10,000 rpm and can be stopped in 4 seconds. Assume the deceleration is uniform. The centrifuge radius is 8 cm.
(i)What is the average angular acceleration of the centrifuge? (2%)
(ii)What is the distance that a point on the rim travels during deceleration?
angular acceleration=10,000*2PI/60 / 4 rad/sec
displacement=wi*t -1/2 10,000*2PI/60 * 4^2
To determine the answers, we can use the following formulas:
(i) The average angular acceleration (α_avg) is given by the formula:
α_avg = Δω / Δt
(ii) The distance traveled by a point on the rim (s) is given by the formula:
s = r × θ
Where:
Δω is the change in angular velocity
Δt is the change in time
r is the radius of the centrifuge
θ is the angle moved by a point on the rim
Given:
Maximum rotation rate (ω_max) = 10,000 rpm
Time taken to stop (Δt) = 4 seconds
Centrifuge radius (r) = 8 cm
(i) To find the average angular acceleration (α_avg), we first need to find the change in angular velocity (Δω):
Δω = ω_max - 0
= 10,000 rpm - 0
= 10,000 rpm
Converting rpm to rad/s:
1 rpm = (2π rad) / (60 s)
10,000 rpm = (10,000 × 2π) / 60 rad/s
= (20,000π) / 60 rad/s
= 1,000π / 3 rad/s
Now, we can plug the values into the formula for α_avg:
α_avg = Δω / Δt
= (1,000π / 3 rad/s) / 4 s
= 250π / 3 rad/s²
≈ 261.8 rad/s²
Therefore, the average angular acceleration of the centrifuge is approximately 261.8 rad/s².
(ii) To find the distance traveled by a point on the rim (s), we need to find the angle moved by a point on the rim (θ):
Using the formula for angular acceleration:
α = Δω / Δt
We can rearrange it to solve for Δω:
Δω = α × Δt
Plugging in the values:
Δω = (250π / 3 rad/s²) × 4 s
= (1,000π / 3) rad/s
Now, using the formula for distance traveled:
s = r × θ
We can rearrange it to solve for θ:
θ = Δω / r
Plugging in the values:
θ = (1,000π / 3) rad / 8 cm
= (125π / 3) rad/cm
Finally, we can calculate the distance traveled (s):
s = r × θ
= 8 cm × (125π / 3) rad/cm
= (1,000π / 3) cm
≈ 1,047.2 cm
Therefore, the distance that a point on the rim travels during deceleration is approximately 1,047.2 cm.
To find the average angular acceleration of the centrifuge, we can use the formula:
Average angular acceleration = (change in angular velocity) / (change in time)
The change in angular velocity can be calculated by converting the maximum rotation rate from revolutions per minute (rpm) to radians per second (rad/s). There are 2π radians in one revolution and 60 seconds in one minute, so we can use the following conversion factor:
1 rpm = (2π rad) / (60 s)
Thus, the maximum rotation rate of 10,000 rpm is equivalent to:
(10,000 rpm) * (2π rad / 60 s) = (10,000 * 2π) / 60 rad/s
Now, we need to find the change in time for the deceleration. Given that the centrifuge can be stopped in 4 seconds, the change in time is simply 4 s.
Substituting the values into the formula, we have:
Average angular acceleration = [(10,000 * 2π) / 60 rad/s] / 4 s
Simplifying this expression gives us the average angular acceleration.
For part (ii), we need to find the distance that a point on the rim travels during deceleration. This can be calculated using the formula for linear distance or arc length:
Distance = (angular displacement) x (radius)
The angular displacement can be found using the formula:
Angular displacement = (angular velocity) x (time)
In this case, since we are interested in the deceleration, the angular velocity can be taken as the maximum rotation rate.
Substituting the given values into the formula, we have:
Angular displacement = (10,000 * 2π) / 60 rad/s * 4 s
Finally, we multiply the angular displacement by the radius of the centrifuge to find the distance traveled by a point on the rim during deceleration. The radius given is 8 cm, but it is often better to use consistent units, so let's convert it to meters before the calculation:
Distance = (angular displacement) x (radius) = ([(10,000 * 2π) / 60 rad/s] * 4 s) * (8 cm / 100 cm/m) = [(10,000 * 2π) / 60 rad/s] * 4 s * (8 / 100) m
Simplifying this expression gives us the distance traveled by a point on the rim during deceleration.