The angular velocity of a wheel is given by w = 3t^2 + 7t.
a. What is the wheel’s initial angular velocity at t = 0?
I think this should be zero but that seems too simple?
b. What is the instantaneous angular acceleration (α) at t = 2 sec?
Do I need to take the derivative of this as a function (6t + 7) and substitute 2 for t to get instantaneous acceleration?
c. What is the wheel’s angular displacement from t = 1 sec to t = 3 sec?
I need to know w to solve for angular displacement...what should w be? I thought it was zero but that doesn't seem right
Am I on the right track here?
a, yes simple
b. yes correct.
c. Just integrate displacement from t=1 to 3
displacement= INT w(t) dt over limits. w(t)= 3t^2 + 7t.
Thank you!
a. Yes, you are correct. The initial angular velocity represents the angular velocity at t = 0. To find it, you can substitute t = 0 into the given equation for angular velocity:
w = 3t^2 + 7t
w = 3(0^2) + 7(0)
w = 0 + 0
w = 0
Therefore, the wheel's initial angular velocity at t = 0 is indeed 0.
b. For the instantaneous angular acceleration at t = 2 seconds, you need to take the derivative of the angular velocity function with respect to time, and then substitute t = 2. The derivative of the given angular velocity function w = 3t^2 + 7t with respect to t gives you the equation for angular acceleration:
α = dw/dt = d(3t^2 + 7t)/dt
α = 6t + 7
Now, substitute t = 2 into the equation to find the instantaneous angular acceleration at t = 2:
α = 6(2) + 7
α = 12 + 7
α = 19
Therefore, the instantaneous angular acceleration at t = 2 seconds is 19.
c. To find the wheel's angular displacement from t = 1 sec to t = 3 sec, you would need to integrate the angular velocity function from t = 1 to t = 3. However, you have not provided any information about the initial condition or the limits of integration, which would be necessary to determine the value of angular displacement accurately. The angular displacement is not solely determined by the angular velocity function w = 3t^2 + 7t. Therefore, without knowing the initial condition or the limits of integration, it is not possible to determine the wheel's angular displacement from t = 1 sec to t = 3 sec.
To recap:
a. The wheel's initial angular velocity at t = 0 is zero.
b. The instantaneous angular acceleration at t = 2 sec is 19.
c. Without additional information, it is not possible to determine the wheel's angular displacement from t = 1 sec to t = 3 sec.