A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?

Answer 1

I got: #2.75xx10^4#revolutions
BUT check my maths!

We can say that: #1"rpm"=(2pi)/60(rad)/s#
[1 revolution is #2pi# radians and 1 min=60 s]
In our case the object changes ANGULAR velocity and goes from #omega_0=0# to #omega_f=15,000"rpm"=15,000xx(2pi)/60(rad)/s# in #t=220s#. This corresponds to an angular acceleration: #alpha=(omega_f-omega_0)/t=15,000xx(2pi)/60xx1/220=7.14(rad)/s^2# Let us use a (rotational) kinematic relationship to find tha ANGULAR distance #theta# described, as: #theta=omega_0t+1/2alphat^2# #theta=0+1/2*7.14*(220^2)~~1.72xx10^5rad# in total (during the #220s#)
But each revolution corresponds to #2pi# rad giving us: #(1.7xx10^5)/(2pi)=2.75xx10^4#revolutions
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Answer 2

To calculate the number of revolutions, we first need to convert the final angular velocity from rpm to revolutions per second (rev/s). Then, we can use the formula for angular displacement to find the total number of revolutions.

Given: Initial angular velocity (ωi) = 0 rpm Final angular velocity (ωf) = 15,000 rpm Time (t) = 220 s

Convert final angular velocity to rev/s: ωf = (15,000 rpm) * (1 min / 60 s) * (1 rev / 1 rpm) = 250 rev/s

Use the formula for angular displacement: θ = ωi * t + (1/2) * α * t^2

Since the centrifuge starts from rest, ωi = 0.

θ = (1/2) * α * t^2 α = (ωf - ωi) / t α = (250 rev/s - 0 rev/s) / 220 s = 1.136 rev/s^2

θ = (1/2) * (1.136 rev/s^2) * (220 s)^2 θ ≈ 13,726 rev

Therefore, the centrifuge turned approximately 13,726 revolutions in 220 seconds.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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