Respuesta :
Answer:
number of revolutions = 5833.333
Explanation:
The three equations for uniformly accelerated straight line motion are,
v = u + at
s = ut + [tex]\frac{1}{2} at^{2}[/tex]
[tex]v^{2} = u^{2} + 2as[/tex]
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 + [tex]\frac{1}{2}[/tex]α[tex]t^{2}[/tex]
ω[tex]^{2}[/tex] = ω₀[tex]^{2}[/tex] + 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 [tex]\frac{2\pi}{60}[/tex] radian/sec = [tex]\frac{70000\pi }{6}[/tex]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 = [tex]\frac{\frac{70000\pi }{6}-0}{2}[/tex]
α = [tex]\frac{70000\pi }{12}[/tex] radian/[tex]sec^{2}[/tex]
We know,
ω[tex]^{2}[/tex] = ω₀[tex]^{2}[/tex] + 2αθ
rearranging to find θ
θ = (ω^2 - ω₀^2)/2α
= [tex]\frac{(\frac{70000\pi }{6})^{2} - 0^{2} }{2(\frac{70000\pi }{12})}[/tex]
= [tex]\frac{70000\pi }{6}[/tex] 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 = [tex]\frac{\frac{70000\pi }{6}}{2\pi }[/tex] revolutions
= [tex]\frac{70000}{12}[/tex]revolutions = 5833.333 revolutions
