You are designing a centrifuge to spin at a rate of 13,880 rev/min.
(a) Calculate the maximum centripetal acceleration that a test-tube sample held in the centrifuge arm 14.6 cm from the rotation axis must withstand.
(b) It takes 1 min, 17 s for the centrifuge to spin up to its maximum rate of revolution from rest. Calculate the magnitude of the tangential acceleration of the centrifuge while it is spinning up, assuming that the tangential acceleration is constant.
(a) To calculate the maximum centripetal acceleration, we can use the formula:
Centripetal acceleration (a) = (angular velocity)^2 x radius
Given:
Angular velocity (ω) = 13,880 rev/min
First, let's convert the angular velocity to radians per second:
1 rev = 2π radians
So, the angular velocity in radians per second is:
ω = 13,880 rev/min x (2π radians/1 rev) x (1 min/60 s) = 1454.13 rad/s
Now, let's calculate the maximum centripetal acceleration:
Radius (r) = 14.6 cm = 0.146 m
a = (1454.13 rad/s)^2 x 0.146 m = 301,179.91 m/s^2
Therefore, the maximum centripetal acceleration that the test-tube sample must withstand is 301,179.91 m/s^2.
(b) To calculate the magnitude of the tangential acceleration while the centrifuge is spinning up, we can use the formula:
Tangential acceleration (a_t) = (change in tangential velocity) / (time taken)
Given:
Time taken (t) = 1 min, 17 s = 77 s
Change in tangential velocity (Δv) = maximum tangential velocity (v) - initial tangential velocity (v_0)
We are assuming that the tangential acceleration is constant, so the change in tangential velocity is:
Δv = v - v_0
v_0 = 0 m/s (since the centrifuge is starting from rest)
v = (angular velocity) x (radius)
Using the values from part (a):
v = 1454.13 rad/s x 0.146 m = 212.19 m/s
Δv = 212.19 m/s - 0 m/s = 212.19 m/s
Now, let's calculate the magnitude of the tangential acceleration:
a_t = Δv / t = 212.19 m/s / 77 s = 2.755 m/s^2
Therefore, the magnitude of the tangential acceleration of the centrifuge while it is spinning up is 2.755 m/s^2.
To calculate the maximum centripetal acceleration and the tangential acceleration of the centrifuge, we can use the following formulas:
(a) The maximum centripetal acceleration (ac) of an object moving in a circle can be calculated using the following formula:
ac = (ω^2) * r,
where ω is the angular velocity and r is the radius of rotation.
Given that the centrifuge spins at a rate of 13,880 rev/min, we need to convert this to radians per second. Since 1 revolution is equal to 2π radians, we have:
ω = (13,880 rev/min) * (2π rad/rev) / (60 s/min),
After converting the units, we get:
ω = 1457.67 rad/s.
The radius of the test-tube sample from the rotation axis is given as 14.6 cm. Converting this to meters, we have:
r = 14.6 cm * (1 m/100 cm),
Thus,
r = 0.146 m.
Now we can substitute the values into the formula to calculate the maximum centripetal acceleration:
ac = (1457.67 rad/s)^2 * 0.146 m.
Calculating it will give you the value of the maximum centripetal acceleration.
(b) To calculate the tangential acceleration, we need to use the following formula:
at = (ω * Δt) / 2π,
where ω is the angular velocity and Δt is the time taken for the centrifuge to spin up to its maximum rate of revolution.
The angular velocity ω is already calculated as 1457.67 rad/s.
Δt is given as 1 min and 17 s, which can be converted to seconds:
Δt = 1 min * 60 s/min + 17 s.
Substituting the values into the formula, we can calculate the tangential acceleration.
a) 8.68 m/s^2
b) 0.000766 m/s^2