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 answer these questions, we need to understand the concept of angular acceleration and the equations related to it. Angular acceleration is the rate at which an object's angular velocity changes with respect to time.
a) To calculate the angular acceleration of the turbine blade, we'll use the following formula:
α = (ωf - ωi) / t
Where:
- α is the angular acceleration
- ωf is the final angular velocity
- ωi is the initial angular velocity
- t is the time taken
Given:
ωi = 3850 rpm (convert it to rad/s by multiplying by 2π/60)
ωf = 0 rpm (which is 0 rad/s)
t = 3.8 seconds
Plugging in the values:
α = (0 - (3850 * 2π/60)) / 3.8
Calculate α using the given values to get the angular acceleration of the turbine blade.
b) To calculate the distance travelled by a point on the rim of the turbine blade during the acceleration, we can use the formula for angular displacement:
θ = ωi * t + (1/2) * α * t^2
Where:
- θ is the angular displacement
- ωi is the initial angular velocity
- t is the time taken
- α is the angular acceleration
Given:
ωi = 3850 rpm (convert it to rad/s by multiplying by 2π/60)
α (from part a)
t = 3.8 seconds
Plugging in the values:
θ = (3850 * 2π/60) * 3.8 + (1/2) * α * (3.8)^2
Calculate θ using the given values to get the angular displacement of a point on the rim of the turbine blade during the acceleration.
c) The magnitude of the net displacement of a point on the rim of the turbine blade during the deceleration can be calculated using the formula:
d = r * θ
Where:
- d is the displacement
- r is the radius of the turbine blade (given as 65 cm which is 0.65 m)
- θ is the angular displacement (from part b)
Plugging in the values:
d = 0.65 * θ
Calculate d using the given values to get the magnitude of the net displacement of a point on the rim of the turbine blade during the deceleration.
Remember to convert units as necessary and perform the calculations to get the final answers for parts a, b, and c.