Question

A clown pedals a unicycle applying an average torque of 7.50 Nm to its wheel. The wheel has a mass of 2.50 kg with an outside diameter of 0.600 m and an inside diameter of 0.525 m. The unicyclist and the unicycle have a combined mass of 90.0 kg. What is his translational and tangential acceleration? Be careful! Only the wheel's mass rotates but the total mass accelerates horizontally. [Answer: 0.285 m/s2]

Answer 1

Task:

A clown pedals a unicycle applying an average torque of $7.50 \, \text{Nm}$ to its wheel. The wheel has a mass of $2.50 \, \text{kg}$ with an outside diameter of $0.600 \, \text{m}$ and an inside diameter of $0.525 \, \text{m}$ . The unicyclist and the unicycle have a combined mass of $90.0 \, \text{kg}$ . What is his translational and tangential acceleration? Be careful! Only the wheel's mass rotates but the total mass accelerates horizontally. [Answer: $0.285 \, \text{m/s}^2$ ]

Solution:

$$\begin{array}{l} \tau_ {n e t} = \frac {d L}{d t} = \frac {d I \omega}{d t} = \frac {I d \omega}{d t} = I \cdot \alpha = 7. 5 N m \\ I = \frac {1}{8} \left(m _ {\text {w h e e l}} + m _ {\text {u n i c y c l i s t}}\right) \left(0. 6 ^ {2} m ^ {2} + 0. 5 2 5 ^ {2} m ^ {2}\right) = \frac {1}{8} \cdot 9 0 k g \cdot 0. 6 3 5 6 2 5 m ^ {2} = \\ = 7. 1 5 0 7 8 1 2 5 k g \cdot m ^ {2} \\ \alpha = 0. 9 5 3 4 3 7 5 \frac {r a d}{s ^ {2}} \\ \alpha = \frac {d \omega}{d t} = \frac {d \frac {v}{0 . 3}}{d t} = \frac {1}{0 . 3} \frac {d v}{d t} = \frac {1}{0 . 3} a _ {t a n} \\ \end{array}$$ $$a_{tan} = 0.28603125 \frac{m}{s^{2}}$$ $$a_{tan} = a = 0.28603125 \frac{m}{s^{2}}$$

Answer:

$$a = 0.28603125 \frac{m}{s^{2}}$$