The angular velocity of a wheel is given by w(t) = (2.00 rad/s^2)t + (1.00 rad/s^4)t^3.
A. What is the angular displacement of the wheel from time t=0.00s to time t=T?
B.What is the angular acceleration of the wheel as a function of time?
To answer these questions, we need to understand the concepts of angular displacement and angular acceleration.
Angular displacement is a measure of how much an object has rotated or turned around a fixed axis. It is usually measured in radians (rad) or degrees (°). In this case, the angular displacement is given by the integral of the angular velocity function over the given time interval.
Angular acceleration, on the other hand, is the rate at which the angular velocity of an object changes with time. It is the derivative of the angular velocity function.
Now, let's find the answers to the specific questions:
A. What is the angular displacement of the wheel from time t=0.00s to time t=T?
To find the angular displacement, we need to integrate the angular velocity function over the given time interval (from t=0.00s to t=T).
The given angular velocity function is:
w(t) = (2.00 rad/s^2)t + (1.00 rad/s^4)t^3
To find the angular displacement, we integrate the angular velocity function:
Θ(t) = ∫[0 to T] w(t) dt
Integrating the given function, we get:
Θ(t) = (1.00 rad/s^2)t^2 + (0.25 rad/s^4)t^4 | [0 to T]
Plugging in the upper limit (T) and lower limit (0), we get:
Θ(T) = (1.00 rad/s^2)(T)^2 + (0.25 rad/s^4)(T)^4 - [0]
Therefore, the angular displacement of the wheel from time t=0.00s to time t=T is (1.00 rad/s^2)(T)^2 + (0.25 rad/s^4)(T)^4.
B. What is the angular acceleration of the wheel as a function of time?
The given angular velocity function is:
w(t) = (2.00 rad/s^2)t + (1.00 rad/s^4)t^3
To find the angular acceleration, we take the derivative of the angular velocity function with respect to time (t).
α(t) = dw(t)/dt
Differentiating the given function, we get:
α(t) = (2.00 rad/s^2) + (3.00 rad/s^4)t^2
Therefore, the angular acceleration of the wheel as a function of time is (2.00 rad/s^2) + (3.00 rad/s^4)t^2.