A wheel of radius 0.5 m rolls without sliding on a horizontal surface as shown. Starting from rest, the wheel moves with constant angular acceleration 6 rad/s². The distance in traveled by the center of the wheel from t = 0 to t = 3 s is: A) 0 m B) 27 m C) 13.5 m D) 18 m E) none of these

Question

Quizplus · Accepted Answer

We start with the kinematic equation for translational motion: 
$$x(t) = \frac{1}{2}at^2+v_0t+x_0$$
where $x(t)$ is the position at time $t$, $a$ is the acceleration, $v_0$ is the initial velocity, and $x_0$ is the initial position.

In this case, the translational motion is the motion of the center of the wheel. We will use the fact that for rolling motion without slipping, the translational acceleration is related to the rotational acceleration by $a = R\alpha$, where $R$ is the radius of the wheel and $\alpha$ is the angular acceleration. In this case, $R = 0.5	ext{ m}$ and $\alpha = 6	ext{ rad/s}^2$.

At $t=0$, the wheel is at rest, so $v_0=0$ and $x_0=0$. Plugging in $a = R\alpha$, we get:
$$x(t) = \frac{1}{2}R\alpha t^2 + x_0$$
Plugging in the values, we get:
$$x(3	ext{ s}) = \frac{1}{2}(0.5	ext{ m})(6	ext{ rad/s}^2)(3	ext{ s})^2 + 0	ext{ m} = 13.5	ext{ m}$$
Therefore, the answer is (C) 13.5 m.

A Wheel of Radius 0