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Question

# An athlete in a gym applies a constant force of 50 N to the pedals of a bicycle to keep the rotation rate of the wheel at 10 rev/s. The length of the pedal arms is 30 cm. What is the power delivered to the bicycle by the athlete?

Solution

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We know that :

$\tau = r_{\perp} F$

And the power :

$P = \tau \omega$

From $\textbf{the kinematics of the rotational motion }$ we know that :

$\left[ \dfrac{10 \times 60\; \text {rev}}{\text{min}} \cdot \dfrac{2 \pi\; \text{rad}}{\text{rev}} \cdot \dfrac{1 \text{min}}{60\; \text{s}} = 62.83\; \mathrm{rad/s} \right]$

Where :

• $F$ is the force acting on the pedals .
• $r$ is the radius of the pedals .
• $\omega$ is the angular velocity of the pedals .
• $\tau$ is the torque acting on the pedals due to the force .

$\textbf{Givens}$ : $r_{\perp} = 0.3 \ \text{m}$ , $F= 50 \ \mathrm{N}$ and $\omega = 62.83\ \mathrm{rad/s}$.

$\textbf{Plugging }$ known information into to get :

\begin{align*} \tau & = r_{\perp} F \\ &= 0.3 \times 50 \\ &= 15 \\ P &= \tau \omega \\ &= 15 \times 62.83\\ &=942.45 \end{align*}

$\boxed{P = 942.45 \ \text{W} }$

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