14. Torque & Rotational Dynamics

If you increase the length of the lever arm ($r$) while applying force ($F$) to turn a wrench, how does the amount of force required to produce the same torque ($\tau$) change? (Recall: $\tau =rF$.)

A pulley of radius $R$ and moment of inertia $I$ has two ropes wrapped around it on opposite sides. The tensions in the ropes are $12 N$ and $10 N$. What is the angular acceleration $\alpha$ of the pulley?

Which equation correctly expresses the relationship between $\tau$ (torque) and $\alpha$ (angular acceleration) for a rigid body rotating about a fixed axis?

Which of the following is an effective way to increase the amount of $\tau$ (torque) produced by a wind turbine?

Suppose that piano has a long, thin bar ran through it (totally random), shown below as the vertical red line, so that it is free to rotate about a vertical axis through the bar. You push the piano with a horizontal 100 N (blue arrow), causing it to spin about its vertical axis with 0.3 rad/s² . Your force acts at a distance of 1.1 m from the bar, and is perpendicular to a line connecting it to the bar (green dotted line). What is the piano's moment of inertia about its vertical axis?