Question

the st joseph river swing bridge in st joseph, michigan has a mass of 300 tons (2.72×10⁵ kg) and is 231 ft(70.4 m) long. if the motor produces 563 knm of torque and takes 10 s to accelerate the bridge to 0.05 rad/s, what is the bridge’s moment of inertia?

Explanation:

Step1: Calculate angular acceleration

First, find angular acceleration $\alpha$ using $\alpha = \frac{\Delta \omega}{\Delta t}$.
$\alpha = \frac{0.05\ \text{rad/s} - 0}{10\ \text{s}} = 0.005\ \text{rad/s}^2$

Step2: Convert torque to SI units

Convert torque $\tau$ from kNm to Nm:
$\tau = 563\ \text{kNm} = 563 \times 10^3\ \text{Nm}$

Step3: Solve for moment of inertia

Use rotational dynamics formula $\tau = I\alpha$, rearrange to $I = \frac{\tau}{\alpha}$.
$I = \frac{563 \times 10^3\ \text{Nm}}{0.005\ \text{rad/s}^2}$

Answer:

$1.126 \times 10^8\ \text{kg·m}^2$