Question 5: Rolling Cone Kinematics
The toy acts as a conical frustum. For rolling without slipping, the apex of the cone must be stationary on the ground. The toy rotates about the vertical axis passing through this apex \( O \).
Let \( \theta \) be the semi-vertical angle of the cone: \[ \tan\theta = \frac{r_2 – r_1}{l} = \frac{10.05 – 9.95}{1.00} = 0.1 \]
The center of the rod is at the mean radius \( r_{avg} = \frac{r_1 + r_2}{2} = 10.0 \) cm. The horizontal distance \( d \) from the center of the rod to the vertical axis is: \[ d = \frac{r_{avg}}{\tan\theta} = \frac{10.0}{0.1} = 100 \text{ cm} \]
The center moves with speed \( v_c = 10.0 \) cm/s in a circle of radius \( d \). The angular velocity \( \Omega \) about the vertical axis is: \[ \Omega = \frac{v_c}{d} = \frac{10.0}{100} = 0.1 \text{ rad/s} \]