A70 cm diameter wheel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating

A) Angular acceleration = 3.93 rad/s²

B) i) Tangential components = 2.75 m/s²

   ii) Radial components = 323.6 m/s²

Given data :

Diameter of wheel = 70 cm = 0.7 m

Initial angular frequency ( f₁ ) = 130 rpm = 2.17 Hz

Final angular frequency ( f₂ ) = 280 rpm = 4.67 Hz

Time  ( T ) = 4 secs

A) Calculate the angular acceleration

Angular acceleration ( ∝ )  =  Δω / ΔT

                                             = 15.71 / 4  = 3.93 rad/s²

where : Δω = 2π ( f₂ - f₁ ) = 2π ( 4.67 - 2.17 ) = 15.71

         change in time ( ΔT ) = 4 secs

B) Determine the radial and tangential components of the linear acceleration after 2 secs

First step : calculate the angular speed after 2 secs

ωf = ωf₁ + ∝t

    = 2π ( 2.17 ) + 3.93 * 2

    = 21.5 rad/sec

Final step :

i) Tangential acceleration

 at = r∝

 at = 0.70 ( 3.93 ) = 2.75 m/s²

ii ) Radial acceleration

  ar = ω²r

      = ( 21.5 )² * ( 0.70 )  = 323.6 m/s²

Hence we can conclude that A) Angular acceleration = 3.93 rad/s²

B) i) Tangential components = 2.75 m/s², ii) Radial components = 323.6 m/s²

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Part a)

as we know that angular speed is given as

here we have

now for angular acceleration we have

Part b)

angular speed of the wheel after t = 2 s

now we have

tangential acceleration

radial acceleration will be

the part of the wave he is travelling on is the crest