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 calculate the angular acceleration of the turbine blade, we can use the formula:
Angular acceleration = (Final angular velocity - Initial angular velocity) / Time
a) Angular acceleration = (0 rpm - 3850 rpm) / 3.8 seconds
To convert rpm to radians per second, we multiply by 2π/60:
Angular acceleration = (0 - 3850) * (2π/60) / 3.8 seconds
b) To find the distance traveled by a point on the rim of the turbine blade during acceleration, we need to calculate the angular displacement. The formula to calculate angular displacement is:
Angular displacement = (1/2) * angular acceleration * (time)^2
The distance traveled by a point on the rim of the turbine blade during acceleration is given by:
Distance = (Angular displacement) * (radius of the turbine blade)
The radius of the turbine blade is half of its diameter, which is 65 cm / 2 = 32.5 cm = 0.325 m.
c) To find the net displacement of a point on the rim of the turbine blade during deceleration, we can use the kinematic equation:
Displacement = (Initial velocity * Time) + (1/2) * acceleration * (Time)^2
Since the turbine blade slows down to 0 rpm, the initial velocity is 3850 rpm. To convert rpm to radians per second, we multiply by 2π/60:
Initial velocity = 3850 rpm * (2π/60)
Net displacement = (Initial velocity * Time) + (1/2) * acceleration * (Time)^2
Let's calculate these values:
a) Angular acceleration = (0 - 3850) * (2π/60) / 3.8 seconds
b) Angular displacement = (1/2) * angular acceleration * (time)^2
Distance = Angular displacement * radius of the turbine blade
c) Initial velocity = 3850 rpm * (2π/60)
Net displacement = (Initial velocity * Time) + (1/2) * acceleration * (Time)^2
To find the angular acceleration of the turbine blade, you need to know the change in angular velocity and the time taken. The formula to calculate angular acceleration is:
Angular acceleration (α) = (change in angular velocity) / (time taken)
a) The change in angular velocity can be calculated by subtracting the final angular velocity (0 rpm) from the initial angular velocity (3850 rpm). However, we need to convert the units from rpm (revolutions per minute) to radians per second (rad/s) before proceeding with the calculation.
1 revolution = 2π radians
Therefore, 3850 rpm = (3850 * 2π) rad/min
To convert it to rad/s, divide by 60 (since there are 60 seconds in a minute):
Change in angular velocity = (3850 * 2π) / 60 rad/s
Now, substitute the values into the formula:
Angular acceleration (α) = [(3850 * 2π) / 60] / 3.8 rad/s²
To find the distance traveled by a point on the rim of the turbine blade during the acceleration, you need to know the initial angular velocity, time taken, and radius of the turbine blade. A point on the rim of the blade travels a distance equal to the circumference of the circle.
b) The circumference of a circle can be calculated using the formula:
Circumference = 2π * radius
The radius of the turbine blade is half its diameter, which is given as 65 cm. Therefore, the radius (r) is 65 cm / 2 = 32.5 cm = 0.325 m.
Now, substitute the values into the formula:
Distance traveled by a point on the rim = (Initial angular velocity) * (time taken) * (radius)
Distance traveled = [(3850 * 2π) / 60] * 3.8 * 0.325 m
To find the magnitude of the net displacement of a point on the rim of the turbine blade during the deceleration, you need to use the same formula as above with the final angular velocity being 0 rpm.
c) Substitute the values into the formula:
Magnitude of net displacement = (Final angular velocity) * (time taken) * (radius)
Magnitude of net displacement = 0 * 3.8 * 0.325 m
Since the final angular velocity is 0, the magnitude of the net displacement during deceleration is 0 meters.