In a power plant the high pressure turbine trips due to an unauthorised valve shutdown. What are
a) what is the angular acceleration of the turbine blade?
b) what is the distance travelled by a point on the rim of the turbine blade during the acceleration?
c) what is the magnitude of the net displacement of a point on the rim of the turbine blade during the deceleration?
If the turbines 65 cm diameter blades slow down from 3850 rpm to 0 rpm in 3.8 seconds.
To solve these questions, we can use some basic formulas related to rotational motion. Let's start with the given information:
Diameter of the turbine blades (d) = 65 cm = 0.65 m
Initial angular velocity (ω_i) = 3850 rpm
Final angular velocity (ω_f) = 0 rpm
Time taken (t) = 3.8 seconds
Now, let's proceed with the step-by-step solution:
a) To find the angular acceleration of the turbine blade, we can use the formula:
Angular acceleration (α) = (ω_f - ω_i) / t
Substituting the given values, we have:
α = (0 - 3850 rpm) / 3.8 seconds
α = (-3850 rpm) / 3.8 seconds (Note: To convert rpm to rad/s, multiply by 2π/60)
α ≈ -641.053 rad/s² (rounded to three decimal places)
Therefore, the angular acceleration of the turbine blade is approximately -641.053 rad/s².
b) To calculate the distance traveled by a point on the rim of the turbine blade during acceleration, we can use the formula:
Distance (s) = ω_i * t + (1/2) * α * t²
Substituting the given values, we have:
s = 3850 rpm * 3.8 seconds + (1/2) * (-641.053 rad/s²) * (3.8 seconds)²
s ≈ 5918.632 m (rounded to three decimal places)
Therefore, the distance traveled by a point on the rim of the turbine blade during acceleration is approximately 5918.632 meters.
c) To calculate the net displacement of a point on the rim of the turbine blade during deceleration, we need to calculate the final angular displacement (θ_f) using the formula:
θ_f = ω_i * t + (1/2) * α * t²
Substituting the given values, we have:
θ_f = 3850 rpm * 3.8 seconds + (1/2) * (-641.053 rad/s²) * (3.8 seconds)²
θ_f ≈ 5918.632 radians (rounded to three decimal places)
Since the turbine blade is coming to a stop, the deceleration will cause the point on the rim to rotate in the opposite direction, and the net displacement will be equal to the magnitude of the final displacement. Therefore,
Magnitude of net displacement ≈ 5918.632 radians.
Note: The surface distance traveled and the net displacement are the same in this case, as there is no change in direction during deceleration.
Please note that these calculations are based on the given information and assumptions made about the motion of the turbine blade.
To find the answers to these questions, we need to use some basic physics equations related to angular acceleration, distance, and displacement.
a) Angular acceleration (α) can be calculated using the equation:
α = (ω2 - ω1) / t
where:
ω2 = final angular velocity (0 rpm in this case, which we can convert to radians per second)
ω1 = initial angular velocity (3850 rpm, which we can convert to radians per second)
t = time taken for the change in angular velocity (3.8 seconds)
To convert rpm to radians per second, we use the conversion factor: 1 revolution = 2π radians.
First, we convert the initial and final angular velocities to radians per second:
ω1 = 3850 rpm * (2π radians/1 revolution) * (1 minute/60 seconds) = ω1 in radians per second
ω2 = 0 rpm * (2π radians/1 revolution) * (1 minute/60 seconds) = ω2 in radians per second
Substituting the values into the equation, we get:
α = (0 - ω1) / t
b) The distance traveled by a point on the rim of the turbine blade during the acceleration can be calculated using the equation:
θ = ω1 * t + 0.5 * α * t^2
where:
θ = angle in radians covered by the turbine blade during acceleration
c) The magnitude of the net displacement of a point on the rim of the turbine blade during deceleration will be equal to the distance traveled during acceleration but in the opposite direction.
Now, let's calculate the answers.
First, convert the given diameter of 65 cm to radius (r):
r = diameter / 2 = 65 cm / 2 = 32.5 cm = 0.325 m
a) Calculate Angular Acceleration (α):
Convert ω1 to radians per second:
ω1 = 3850 rpm * (2π radians/1 revolution) * (1 minute/60 seconds) = ω1 in radians per second
Use the equation: α = (0 - ω1) / t
Substitute the values: α = (0 - ω1) / 3.8 seconds
b) Calculate Distance Traveled during Acceleration (θ):
Use the equation: θ = ω1 * t + 0.5 * α * t^2
Substitute the values: θ = ω1 * 3.8 seconds + 0.5 * α * (3.8 seconds)^2
c) Calculate Net Displacement during Deceleration:
The magnitude of the net displacement during deceleration will be the same as the distance traveled during acceleration but in the opposite direction.
Now, you can plug in the values and calculate the answers using the above equations.