An automobile engine slows down from 3500 rpm to 1100 rpm in 4.5 s.
(a) Calculate its angular acceleration, assumed constant.
rad/s2
(b) Calculate the total number of revolutions the engine makes in this time.
rev
I got answer for part a that is -55.81rad/s^2 but it is difficult to find part b
Ans is 172.5 rpm/min
To find the total number of revolutions the engine makes in this time, we need to calculate the change in angular displacement.
We can use the following formula to calculate the change in angular displacement:
Δθ = ωf - ωi
Where:
Δθ = change in angular displacement (in radians)
ωf = final angular velocity (in radians per second)
ωi = initial angular velocity (in radians per second)
Given:
ωi = 3500 rpm = 3500 * 2π / 60 rad/s
ωf = 1100 rpm = 1100 * 2π / 60 rad/s
ωi = 3500 * 2π / 60 = 183.26 rad/s
ωf = 1100 * 2π / 60 = 114.59 rad/s
Now we can calculate the change in angular displacement:
Δθ = 114.59 - 183.26 = -68.67 radians
Since the engine is slowing down, the change in angular displacement is negative.
To convert the change in angular displacement to revolutions, we can use the formula:
1 revolution = 2π radians
So, the total number of revolutions the engine makes in this time can be calculated as:
revolutions = Δθ / (2π)
revolutions = -68.67 / (2π) ≈ -10.93 rev
Note that the negative sign indicates that the engine made 10.93 revolutions in the opposite direction.
Therefore, the total number of revolutions the engine makes in this time is approximately -10.93 revolutions.
To find the total number of revolutions the engine makes in this time, you can follow these steps:
Step 1: Calculate the change in angular velocity (∆ω)
∆ω = final angular velocity - initial angular velocity
Given:
Initial angular velocity (ω₁) = 3500 rpm
Final angular velocity (ω₂) = 1100 rpm
Convert the angular velocities to rad/s:
ω₁ = 3500 rpm * (2π rad/1 min) * (1 min / 60 s) = (3500 * 2π) / 60 rad/s
ω₂ = 1100 rpm * (2π rad/1 min) * (1 min / 60 s) = (1100 * 2π) / 60 rad/s
∆ω = ω₂ - ω₁
Step 2: Calculate the time in seconds
Given: Time (t) = 4.5 s
Step 3: Calculate the average angular velocity (ω_avg)
ω_avg = (∆ω) / t
Step 4: Calculate the total rotation angle (∆θ)
∆θ = ω_avg * t
Step 5: Calculate the total number of revolutions (N)
N = ∆θ / (2π)
Now let's calculate the values.
Given:
ω₁ = (3500 * 2π) / 60 rad/s = 366.52 rad/s
ω₂ = (1100 * 2π) / 60 rad/s = 114.67 rad/s
t = 4.5 s
Step 1:
∆ω = ω₂ - ω₁
∆ω = 114.67 rad/s - 366.52 rad/s
∆ω = -251.85 rad/s
Step 2:
t = 4.5 s
Step 3:
ω_avg = (∆ω) / t
ω_avg = (-251.85 rad/s) / (4.5 s)
ω_avg = -55.87 rad/s
Step 4:
∆θ = ω_avg * t
∆θ = (-55.87 rad/s) * (4.5 s)
∆θ = -251.92 rad
Step 5:
N = ∆θ / (2π)
N = -251.92 rad / (2π)
N ≈ -40.13 revolutions (taking the absolute value)
Since the total number of revolutions cannot be negative, we take the absolute value and the answer for part (b) is approximately 40.13 revolutions.