A flywheel turns through 60 rev as it slows from an angular speed of 2.8 rad/s to a stop.
How much time is required for it to complete the first 30 of the 60 revolutions?
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To find the time required for the flywheel to complete the first 30 revolutions, we need to use the formula:
θ = ω₀t + (1/2)αt²
where:
θ is the angle in radians,
ω₀ is the initial angular velocity in rad/s,
α is the angular acceleration in rad/s²,
t is the time in seconds.
We can rearrange this formula to solve for time:
t = (sqrt(2θ/α) - ω₀) / α
First, let's find the angular acceleration (α).
Angular acceleration (α) can be calculated using the formula:
α = (ω₂ - ω₁) / t
where:
ω₂ is the final angular speed in rad/s,
ω₁ is the initial angular speed in rad/s,
t is the time taken.
Given:
ω₂ = 0 rad/s (since the flywheel comes to a stop),
ω₁ = 2.8 rad/s,
t = ? (we need to find this)
Using the formula for angular acceleration:
α = (0 - 2.8) / t
α = -2.8 / t
Now, let's find the time (t) required for the first 30 revolutions.
θ = 30 * 2π radians (since 1 revolution is equal to 2π radians)
Using the rearranged formula for time:
t = (sqrt(2θ/α) - ω₀) / α
t = (sqrt(2 * 30 * 2π / (-2.8 / t)) - 2.8) / (-2.8 / t)
Simplifying this expression, we get:
t = (sqrt(60πt²/-2.8) - 2.8) / (-2.8 / t)
t = (-sqrt(60πt²/-2.8) + 2.8) / (2.8 / t)
At this point, we can use numerical methods, such as iterative techniques or a graphing calculator, to solve this equation and find the value of t.
ave speed = 1.4 rad/s
stop time = 2 * π * 60 / 1.4
acceleration = 2.8 rad/s / stop time
... negative ... wheel is stopping
30 * 2 * π = 1/2 a t^2 + 2.8 t
... solve for t