Question

un disco accionado por una taladradora gira a 18 000 rpm. cuando deja de accionarse, el disco se detiene después de haber completado 100 vueltas. determina la aceleración angular de frenado y el tiempo que tarda el disco en detenerse. s: 900 π rad/s²; 2/3 s

Explanation:

Step1: Convert angular velocity from rpm to rad/s

The formula to convert from revolutions per minute (rpm) to radians per second (rad/s) is \(\omega = 2\pi n/60\), where \(n = 18000\) rpm.

$$ \omega_{0}=2\pi\times\frac{18000}{60}=600\pi\ \text{rad/s} $$

The final angular velocity \(\omega = 0\) rad/s. The angular displacement \(\theta=100\times2\pi = 200\pi\) rad.

Step2: Find the angular acceleration \(\alpha\)

Use the equation \(\omega^{2}=\omega_{0}^{2}+2\alpha\theta\).
Substitute \(\omega = 0\), \(\omega_{0}=600\pi\) rad/s and \(\theta = 200\pi\) rad into the equation:

$$ 0=(600\pi)^{2}+2\alpha\times(200\pi) $$

$$ - 360000\pi^{2}=400\pi\alpha $$

$$ \alpha=\frac{-360000\pi^{2}}{400\pi}=- 900\pi\ \text{rad/s}^{2} $$

The magnitude of the angular acceleration (since we are asked for the acceleration of braking) is \(900\pi\ \text{rad/s}^{2}\)

Step3: Find the time \(t\)

Use the equation \(\omega=\omega_{0}+\alpha t\).
Substitute \(\omega = 0\), \(\omega_{0}=600\pi\) rad/s and \(\alpha=-900\pi\) rad/s² into the equation:

$$ 0 = 600\pi-900\pi t $$

$$ 900\pi t=600\pi $$

$$ t=\frac{600\pi}{900\pi}=\frac{2}{3}\ \text{s} $$

Answer:

The angular acceleration of braking is \(900\pi\ \text{rad/s}^{2}\) and the time it takes for the disc to stop is \(\frac{2}{3}\ \text{s}\)