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Question

# A computer hard disk starts from rest, then speeds up with an angular acceleration of $\alpha=190 \dfrac{\text{rad}}{\text{s}^2}$ until it reaches its final angular speed of $\omega_f=7200~\text{rpm}$. How many revolutions has the disk made $t=10.0~\text{s}$ after it starts up?

Solution

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Here we are given a final angular speed with the value of:

$\omega_f=7200~\text{rpm}$

Or in $\dfrac{\text{r}}{\text{s}}$:

$\omega_f=7200~\text{rpm}\cdot\frac{1}{60~\text{s}}=120~\frac{\text{r}}{\text{s}}$

The initial angular speed is zero because the hard disk was at the rest at the beginning:

$\omega_i=0~\text{rpm}$

Angular acceleration is given as:

$\alpha=190~\frac{\text{rad}}{\text{s}^2}$

If we know that:

$1~\frac{\text{rad}}{\text{s}}=9.5493~\text{rpm}$

Then our angular acceleration expressed in $\dfrac{\text{rpm}}{\text{s}}$ is:

$\alpha=190\cdot9.5493~\frac{\text{rpm}}{\text{s}}=1814.37~\frac{\text{rpm}}{\text{s}}$

And now in $\dfrac{\text{r}}{\text{s}^2}$ where the previous number we willmultiply with $\dfrac{1}{60~\text{s}}$:

$\alpha=1814.37\cdot\dfrac{1}{60~\text{s}}~\frac{\text{r}}{\text{s}}=30.24~\frac{\text{r}}{\text{s}^2}$

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