The flywheel of a steam engine begins to rotate from rest with a constant angular acceleration of 1.45 rad/s2. It accelerates for 29.5 s, then maintains a constant angular velocity. Calculate the total angle through which the wheel has turned 63.7 s after it begins rotating.
To solve this problem, we will break it down into two parts:
1. Calculating the angle during the acceleration phase
2. Calculating the angle during the constant velocity phase
1. Calculating the angle during the acceleration phase:
During this phase, we have to find the angular displacement covered by the wheel during the first 29.5 seconds.
We can use the formula for angular displacement:
θ = ω_0 * t + 1/2 * α * t^2
Where:
θ is the angular displacement
ω_0 is the initial angular velocity (0 since the wheel starts from rest)
α is the angular acceleration (1.45 rad/s^2)
t is the time interval (29.5 s)
Plugging in the values:
θ = 0 * 29.5 + 1/2 * 1.45 * (29.5)^2
Calculating this expression gives us:
θ = 637.475 rad
2. Calculating the angle during the constant velocity phase:
During this phase, the wheel maintains a constant angular velocity. So we need to find the angular displacement covered during this phase, starting from the end of the acceleration phase (29.5 s) to 63.7 s.
Since the angular velocity is constant, we can use the formula:
θ = ω * t
Where:
θ is the angular displacement
ω is the angular velocity (the value obtained at the end of the acceleration phase)
t is the time interval (63.7 s - 29.5 s)
Plugging in the values:
θ = ω * (63.7 - 29.5)
Now we need to find the value of ω. Since the angular velocity is constant, it is equal to the final angular velocity at the end of the acceleration phase.
Using the formula:
ω = ω_0 + α * t
Where:
ω is the angular velocity
ω_0 is the initial angular velocity (0 since the wheel starts from rest)
α is the angular acceleration (1.45 rad/s^2)
t is the time interval (29.5 s)
Plugging in the values:
ω = 0 + 1.45 * 29.5
Calculating this expression gives us:
ω = 42.775 rad/s
Finally, substituting back into the formula for angular displacement:
θ = 42.775 * (63.7 - 29.5)
Calculating this expression gives us:
θ = 1,383.9475 rad
Therefore, the total angle through which the wheel has turned 63.7 s after it begins rotating is:
Total angle = θ1 + θ2 = 637.475 + 1,383.9475
Total angle = 2,021.4225 rad
To calculate the total angle through which the flywheel has turned, we need to calculate the angular displacement during the first 29.5 seconds of acceleration and the angular displacement during the remaining 63.7 - 29.5 = 34.2 seconds of constant angular velocity.
First, let's calculate the angular displacement during the acceleration phase.
We know that angular acceleration (α) is the rate of change of angular velocity (ω) with respect to time, given as:
α = dω/dt
In this case, α = 1.45 rad/s^2 and the time (t) = 29.5 seconds. We want to find the angular displacement (θ) during this time period.
We can integrate the angular acceleration to get the angular velocity:
∫ dω = ∫ α dt
ω = αt + C
Since the flywheel starts from rest (ω = 0 at t = 0), we can solve for the constant C:
0 = α(0) + C
C = 0
Therefore, the angular velocity during the acceleration phase is given by:
ω = αt
Now, let's calculate the angular displacement during the acceleration phase using the formula:
θ = ωt + (1/2)αt^2
Substituting the values, we have:
θ1 = (1/2)(1.45 rad/s^2)(29.5 s)^2
Next, let's find the angular displacement during the phase of constant angular velocity.
We know that angular velocity (ω) is constant during this phase, and the time (t) is 63.7 seconds. We want to find the angular displacement (θ) during this time period.
The formula for angular displacement during constant angular velocity is:
θ2 = ωt
Since the angular velocity is constant, θ2 = ω(t - t1), where t1 is the duration of the acceleration phase.
Plugging in the values, we have:
θ2 = ω(63.7 s - 29.5 s)
Now, we compute θ2 as:
θ2 = ω(34.2 s)
Finally, we can calculate the total angle through which the wheel has turned by adding the angular displacements during the acceleration and constant velocity phases:
Total angle = θ1 + θ2