Classical Mechanics

Use this for “Blocks of 4 kg and 6 kg hang over a 3 kg disk pulley. Use the Lagrangian to find their acceleration, then find both tensions.”

Use this for “A bead slides on a 0.25 m hoop spun at 9 rad/s about its vertical diameter. At what angle does it rest, and how fast does it rock about that point?”

Use this for “A 0.4 kg bob on a 0.8 m rod passes the bottom at 6 rad/s. Does it swing back or go over the top? If it swings, how far?”

Use this for “A satellite has energy per mass ε = −25 km²/s² and angular momentum per mass h = 55,000 km²/s round Earth. How close and how far does it go?”

Use this for “How fast must a probe move to circle Mars (GM = 4.28 × 10¹³ m³/s²) 3.6 × 10⁶ m from its center, and how fast to escape from there?”

Use this for “A satellite in a circular orbit 7000 km from Earth’s center moves to one at 26,600 km on a Hohmann transfer. Find both burns, the total Δv and the coast time.”

Use this for “A 2 kg tray is 0.5 m by 0.2 m. Find its three principal moments through the center. About which axis will a spin tumble?”

Use this for “A train runs at 80 m/s at latitude 52° N. How big is its sideways Coriolis acceleration, and how long does a Foucault pendulum there take to turn once?”

Use this for “A wall holds a 300 N/m spring, then a 2 kg block, then a 150 N/m spring and a 1 kg block with its far end free. Find both normal-mode frequencies and each mode’s shape.”

Use this for “A car speeds up at 2 m/s². At what angle does a hanging pendulum settle, and what is its period if it is 0.5 m long?”

Use this for “Water turns at 8 rad/s in a bucket. How much higher is it 15 cm from the axis than at the center?”