Question

The drill used by most dentists today is powered by a small air-turbine that can operate at angular speeds of 350000 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 2.0 s. How many revolutions does the drill bit make as it comes up to speed?

Answer 1

number of revolutions = 5833.333

Step-by-step explanation:

The three equations for uniformly accelerated straight line motion are,

v = u + at

s = ut +
`(1)/(2) at^(2)`

`v^(2) = u^(2) + 2as`

where,

v = final velocity

u = initial velocity

s = displacement

a = acceleration

These three equations can be applied to rotational motion also if the angular acceleration is constant just by replacing v with ω, u with ω₀, s with θ, and a with α.

where,

ω = final angular velocity

ω₀ = initial angular velocity

θ = angular displacement

α = angular acceleration

thus, the above equations become,

ω = ω₀ + αt

θ = ut +
`(1)/(2)`α
`t^(2)`

ω
`^(2)` = ω₀
`^(2)` + 2αθ

In the given problem,

initial angular velocity is ω₀ = 0 (since it starts from rest)

final angular velocity is ω = 350000 rpm = 350000 x
`(2\pi)/(60)` radian/sec =
`(70000\pi )/(6)`radian/sec

(since 0ne revolution = 2π radian and one minute = 60 seconds)

time t = 2.0 s

our aim is to find θ. We have two equations with θ. But both of them contains α which is an unknown. So we have to first find α using the first equation.

ω = ω₀ + αt

α = ( ω-ω₀)/t (got by finding α from the above equation.)

Substituting the known values in the equation we get,

α = ( ω-ω₀)/t

α =
`((70000\pi )/(6)-0)/(2)`

α =
`(70000\pi )/(12)` radian/
`sec^(2)`

We know,

ω
`^(2)` = ω₀
`^(2)` + 2αθ

rearranging to find θ

θ = (ω^2 - ω₀^2)/2α

=
`(((70000\pi )/(6))^(2) - 0^(2) )/(2((70000\pi )/(12)))`

=
`(70000\pi )/(6)` radian

but we have to find number of revolutions. We know that one revolution = 2π radian

so to find the number of revolutions we have to divide the answer by 2π.

that is number of revolutions =
`((70000\pi )/(6))/(2\pi )` revolutions

=
`(70000)/(12)`revolutions = 5833.333 revolutions