The gear’s constant angular acceleration α to achieve the angular velocity ω=30 rad/s

Is 3.58 rad/s2

The time required to achieve the angular velocity ω=30 rad/s is 8.38  s .

Given:

The angular acceleration α is 3 rad/s2 .

Final angular velocity ω is 30 rad/s .

The angular displacement of gear θ is 20 rev .

Write the expression of angular displacement for gear.

θ=20 rev(2π rad1 rev)=40 π rad

Write the equation of the angular acceleration.

α=3 rad/s2

Here, angular acceleration is α .

Write the expression for the angular velocity.

ω=ω0+αct (I)

Here, angular velocity is ω , angular acceleration is αc and time is t .

Write the expression for the angular velocity.

ω2=ω02+2αc(θ−θ0) (II)

Here, angular velocity is ω , initial velocity is ω0 , initial revolutions is θ0 , angular acceleration is αc and revolutions is θ .

Conclusion:

Substitute   40π for θ , 0 for θ0 , 0 for ω0 and 30 rad/s for ω in Equation (I).

ω2=ω02+2αc(θ−θ0)(30)2=(0)2+2αc[40π−0]αc=3.581 rad/s2

Thus, the gear’s constant angular acceleration α to achieve the angular velocity ω=30 rad/s is 3.58 rad/s2

Substitute 3.581 rad/s2 for αc , 0 for ω0 and 30 rad/s for ω in Equation (II).

ω=ω0+αct(30)=(0)+(3.581)tt=8.38 s

Thus, the time required to achieve the angular velocity ω=30 rad/s is 8.38  s .