Question

this is a 2 part question22) The drill used by most dentists today is powered by a small air turbine that can operate at angular speeds of 350,000 rpm. These drills, along with ultrasonic dental drills, are the fastest turbines in the world—far exceeding the angular speeds of jet engines. Suppose a drill starts at rest and comes up to operating speed in 2.1 s. (a) Find the angular acceleration produced by the drill, assuming it to be constant. (b) How many revolutions does the drill bit make as it comes up to speed?

Answer 1

(a) The angular acceleration is given by the following formula:

$\alpha=\frac{\omega-\omega_o}{t}$

where,

ωo: initial angular velocity = 0 rpm

ω: final angular velocity = 350,000 rpm

t: time = 2.1 s

convert the given time to minutes:

$2.1s\cdot\frac{1\min}{60s}=0.035\min$

Now, replace the values of the parameters into the formula for the angular acceleration:

$\alpha=\frac{350,000rpm}{0.035\min}=10,000,000\text{ rev/min\textasciicircum{}2}$

The angular acceleration is 10,000,000 revolution per squared minute.

(2) To determine the number of revolutions, use the following fomula:

$\theta=\frac{1}{2}\alpha t^2$

By replacing the values of the parameters, you obtain:

$\theta=\frac{1}{2}(10,000,000)(0.035)^2=6125$

Hence, in a time of 2.1s the drill makes 6125 revolutions.