Which graph best describes the angular position of a wheel with positive constant angular acceleration over time?

The graph that best represents the angular position of a wheel with positive constant angular acceleration over time is a parabola that opens upwards. This is because when an object is undergoing constant angular acceleration, the angular position does not change linearly; instead, it follows a quadratic relationship with time.

Mathematically, angular position (θ) as a function of time (t) under constant angular acceleration (α) can be expressed by the equation:

[

\theta(t) = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2

]

where ( \theta_0 ) is the initial angular position, ( \omega_0 ) is the initial angular velocity, and ( \alpha ) is the constant angular acceleration. The term ( \frac{1}{2} \alpha t^2 ) indicates that the angular position increases more rapidly over time, resulting in a curve that shows a parabolic shape opening upwards. This shape reflects the increasing displacement covered by the wheel due to the constant acceleration, which leads to a larger change in angular position as time progresses.

In contrast, a straight line would suggest a constant angular velocity (not acceleration), a sine wave would