The flywheel of a steam engine runs with a constant angular velocity of 150 rev/min. When steam is shut off, the friction of the bearings and of the air stops the wheel in 2.2 h. (a) What is the constant angular acceleration, in revolutions per minute-squared, of the wheel during the slowdown? (b) How many revolutions does the wheel make before stopping? (c) At the instant the flywheel is turning at 75 rev/min, what is the tangential component of the linear acceleration of a flywheel particle that is 50 cm from the axis of rotation? (d) What is the magnitude of the net linear acceleration of the particle in (c) ?

a) - 1.14 rev/min²

b) 9900 rev

c) - 9.92*10⁻⁴ m/s²

d) 30.8 m/s²

Explanation:

First, convert hours to minutes:

2.2 h * 60 min/h = 132 min

a) Angular acceleration is change in angular velocity over change in time.

α = (ω - ω₀) / t

α = (0 rev/min - 150 rev/min) / 132 min

α = - 1.14 rev/min²

b) θ = θ₀ + ω₀ t + ½ αt²

θ = 0 rev + (150 rev/min) (132 min) + ½ (-1.14 rev/min²) (132 min) ²

θ = 9900 rev

c) The tangential component of linear acceleration is:

a_t = αr

First, convert α from rev/min² to rad/s²:

-1.14 rev/min² * (2π rad/rev) * (1 min / 60 s) ² = - 1.98*10⁻³ rad/s²

Therefore:

a_t = (-1.98*10⁻³ rad/s²) (0.50 m)

a_t = - 9.92*10⁻⁴ m/s²

d) The magnitude of the net linear acceleration can be found from the tangential component and the radial component:

a² = (a_t) ² + (a_r) ²

The radial component is the centripetal acceleration:

a_r = v² / r

a_r = ω² r

First, convert 75 rev/min to rad/s:

75 rev/min * (2π rad/rev) * (1 min / 60 s) = 7.85 rad/s

Find the radial component:

a_r = (7.85 rad/s) ² (0.50 m)

a_r = 30.8 m/s²

Now find the net linear acceleration:

a² = (-9.92*10⁻⁴ m/s²² + (30.8 m/s²) ²

a = 30.8 m/s²