Question

A fan blade rotates with angular velocity given by ωz​(t)=γ−βt2. Part A Calculate the angular acceleration as a function of time. Express your answer in terms of the variables β,γ, and t. Part B If γ=5.45rad/s and β=0.775rad/s3, calculate the instantaneous angular acceleration αz​ at t=2.50 s. Express your answer in radians per second squared. Incorrect; Try Again; 5 attempts remaining Part C If γ=5.45rad/s and β=0.775rad/s3, calculate the average angular acceleration αav−z​ for the time interval t=0 to t=2.50 s. Express your answer in radians per second squared.

Answer 1

Part A: The angular acceleration is -2βt. Part B: The instantaneous angular acceleration at t = 2.50 s is -3.875 rad/s². Part C: The average angular acceleration αav-z for the time interval t = 0 to t = 2.50 s is -3.875 rad/s².

Step-by-step explanation:

Part A: To calculate the angular acceleration, we need to take the derivative of the angular velocity function with respect to time. In this case, the angular velocity function is given as ωz(t) = γ - βt². Taking the derivative, we get αz(t) = -2βt. So the angular acceleration is -2βt.

Part B: To find the instantaneous angular acceleration at t = 2.50 s, we substitute the values γ = 5.45 rad/s, β = 0.775 rad/s³, and t = 2.50 s into the angular acceleration function. αz(t) = -2(0.775)(2.50) = -3.875 rad/s².

Part C: The average angular acceleration αav-z can be found by taking the average of the initial and final angular accelerations over the given time interval. Since the angular acceleration is constant in this case, the average angular acceleration is equal to the instantaneous angular acceleration at any given time during the interval. So the average angular acceleration αav-z is -3.875 rad/s².