A solid sphere slides (no rotation) at v_0 on a rough surface (mu). Its velocity when rolling begins is:

A rod hinged at one end is released from horizontal. The reaction at the hinge when the rod is vertical is:

A solid cylinder is released from rest with a string wound around it, hanging vertically. Its acceleration is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 180 220" style="max-width:180px;margin:0.5em auto;display:block;"> <title>Cylinder hanging from wound string</title> <rect width="180" height="220" fill="#faf9f5" rx="6"/> <defs><marker id="ah-q4s4-04" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0 0, 8 3, 0 6" fill="#c6613f"/></marker></defs> <!-- Support --> <line x1="60" y1="15" x2="120" y2="15" stroke="#14140f" stroke-width="2"/> <!-- String --> <line x1="90" y1="15" x2="90" y2="80" stroke="#14140f" stroke-width="1.5"/> <!-- Cylinder --> <ellipse cx="90" cy="100" rx="35" ry="15" fill="#f0e8e0" stroke="#14140f" stroke-width="1.5"/> <rect x="55" y="100" width="70" height="50" fill="#f0e8e0" stroke="#14140f" stroke-width="1.5"/> <ellipse cx="90" cy="150" rx="35" ry="15" fill="#f0e8e0" stroke="#14140f" stroke-width="1.5"/> <text x="90" y="130" text-anchor="middle" font-size="10" font-family="sans-serif" fill="#14140f">M, R</text> <!-- String connects at edge --> <line x1="90" y1="80" x2="125" y2="100" stroke="#14140f" stroke-width="1.5"/> <!-- T arrow up --> <line x1="130" y1="100" x2="130" y2="65" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q4s4-04)"/> <text x="140" y="80" font-size="9" font-family="sans-serif" fill="#c6613f">T</text> <!-- mg arrow down --> <line x1="90" y1="167" x2="90" y2="200" stroke="#14140f" stroke-width="2" marker-end="url(#ah-q4s4-04)"/> <text x="100" y="195" font-size="9" font-family="sans-serif" fill="#14140f">Mg</text> <!-- a arrow --> <line x1="50" y1="120" x2="50" y2="155" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q4s4-04)"/> <text x="38" y="142" font-size="9" font-family="sans-serif" fill="#c6613f">a=?</text> </svg>

A spinning top (I = 0.01 kg*m^2, omega = 100 rad/s, mass 0.5 kg, CM at 5 cm from pivot) precesses at:

A uniform disc (M, R) has a circular hole of radius R/2 cut out with the hole's centre at R/2 from the disc centre. The MOI about the disc's original central axis is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 230 180" style="max-width:230px;margin:0.5em auto;display:block;"> <title>Disc with hole for MOI calculation</title> <rect width="230" height="180" fill="#faf9f5" rx="6"/> <circle cx="115" cy="90" r="70" fill="#f0e8e0" stroke="#14140f" stroke-width="1.5"/> <circle cx="150" cy="90" r="23" fill="#faf9f5" stroke="#c6613f" stroke-width="1.5" stroke-dasharray="4,3"/> <circle cx="115" cy="90" r="3" fill="#14140f"/> <text x="115" y="105" text-anchor="middle" font-size="8" font-family="sans-serif" fill="#14140f">O</text> <circle cx="150" cy="90" r="2" fill="#c6613f"/> <line x1="115" y1="90" x2="115" y2="20" stroke="#788c5d" stroke-width="1"/> <text x="105" y="55" font-size="8" font-family="sans-serif" fill="#788c5d">R</text> <line x1="150" y1="90" x2="173" y2="90" stroke="#c6613f" stroke-width="1"/> <text x="162" y="85" font-size="7" font-family="sans-serif" fill="#c6613f">R/2</text> <line x1="115" y1="115" x2="150" y2="115" stroke="#14140f" stroke-width="1"/> <text x="132" y="130" text-anchor="middle" font-size="7" font-family="sans-serif" fill="#14140f">R/2</text> <text x="115" y="172" text-anchor="middle" font-size="9" font-family="sans-serif" fill="#14140f">I about O = ?</text> </svg>