Question

A 3.45-kg centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. 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.

(a) What is the angular acceleration (in rad/s^2) of the centrifuge as it spins up?
(b) How many revolutions does the centrifuge make as it goes from rest to its final angular speed?

Answer 1

Answer:

(a) 18.75 rad/s²

(b) 14920.78 rev

Explanation:

(a)

First we find the acceleration of the centrifuge using,

a = (v-u)/t......................... Equation 1

Where v = final velocity, u = initial velocity, t = time.

Given: v = 150 m/s,  u = 0 m/s ( from rest), t = 100 s

Substitute into equation 1

a = (150-0)/100

a = 1.5 m/s²

Secondly we calculate for the angular acceleration using

α = a/r..................... Equation 2

Where α = angular acceleration, r = radius of the centrifuge

Given: a = 1.5 m/s², r = 8 cm = 0.08 m

substitute into equation 2

α = 1.5/0.08

α = 18.75 rad/s²

(b)

Using,

Ф = (ω'+ω).t/2........................... Equation 3

Where Ф = number of revolution of the centrifuge, ω' = initial angular velocity, ω = Final angular velocity.

But,

ω = v/r and ω' = u/r

therefore,

Ф = (u/r+v/r).t/2

where u = 0 m/s (at rest),  = 150 m/s, r = 0.08 m, t = 100 s

Ф = [(0/0.08)+(150/0.08)].100/2

Ф = 93750 rad

If,

1 rad = 0.159155 rev,

Ф = (93750×0.159155) rev

Ф = 14920.78 rev

Answer 2

(a) The angular acceleration of the centrifuge as it spins up is 18.75 rad/s².

(b) The number of revolutions made by the centrifuge is $14,918.8 \ rev$.

The given parameters;

• mass of the centrifuge, m = 3.45 kg
• time of motion, t = 100 s
• radius of the circle, r = 8.0 cm = 0.08 m
• final linear speed, v = 150 m/s

The tangential acceleration of the centrifuge is calculated as follows;

$a = \frac{150 - 0}{100} \\\\a= 1.5 \ m/s^2$

The angular acceleration of the centrifuge as it spins up is calculated as;

$\alpha = \frac{a}{r} \\\\\alpha = \frac{1.5}{0.08} \\\\ \alpha = 18.75 \ rad/s^2$

The number of revolutions made by the centrifuge is calculated as follows;

$\omega _f^2 = \omega _0^2 + 2\alpha \theta \\\\\omega _f^2 = 0 + 2\alpha \theta\\\\\omega _f^2 = 2\alpha \theta\\\\(\frac{150}{0.08} )^2 = 2(18.75)\theta\\\\3,515,625 = 37.5 \theta \\\\\theta = \frac{3,515,625}{37.5} \\\\\theta = 93750 \ radians \\\\\theta =93750 \ rad \times \frac{1 \ rev}{2 \pi \ rad} \\\\\theta = 14,918.8 \ rev$

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