A string is wrapped multiple times around a massive pulley of radius 0.3 meters. A mass is attached at the free end of the string, as shown in the image above. The mass is released at t=0 seconds. Assume friction in the pulley is negligible. This pulley has a moment of inertia of 59 kg·m2. Immediately after t=0, the tension in the rope is 22 newtons. What will be the angular velocity of the pulley after 4 seconds?

Along wire of diameter 1 mm is submerged in an oil bath of temperature ∞ 31°c. the wire has an electrical resistance per unit length of 0.01 ω/m. if a current of 103 a flows through the wire and the convection coefficient is 500 w/m2·k, what is the steady-state temperature of the wire? from the time the current is applied, how long does it take for the wire to reach a temperature that is within 1°c of the steady-state value? the properties of the wire are 8000 kg/m3, 500 j/kg·k, and 20 w/m·k.