Wednesday 6/10 PHYS 2010 Nathalie Hoffmann University of Utah Torque • 𝜏 = 𝑟𝐹𝑠𝑖𝑛𝜙 • r is the distance from the rotation axis to the point where force is applied • F is the magnitude of the force • φ is the angle between the r vector and the F vector • 𝜏 = Ια • Newton’s 2nd Law for rotational motion Problem Solving • Free Body Diagram(s) • Draw all forces • Newton’s 2nd Law • Sum of forces in x direction • Sum of forces in y direction • Sum of torques Angular Momentum • 𝐿 = 𝐼𝜔 • I is moment of inertia • Ω is angular velocity • 𝐿 = 𝑟𝑝𝑠𝑖𝑛𝜙 • r is the distance from rotation axis to particle/object • p is the particle’s/object’s linear momentum • φ is the angle between the r vector and the p vector Practice problems • The drawing shows two identical systems of objects; each consists of the same three small balls connected by massless rods. In both systems the axis is perpendicular to the page, but it is located at a different place, as shown. The same force of magnitude F is applied to the same ball in each system (see the drawing). The masses of the balls are m1 = 9.00 kg, m2 = 6.00 kg, and m3 = 7.00 kg. The magnitude of the force is F = 424 N. • (a) For each of the two systems, determine the moment of inertia about the given axis of rotation. • (b) Calculate the torque (magnitude and direction) acting on each system. • (c) Both systems start from rest, and the direction of the force moves with the system and always points along the 4.00-m rod. What is the angular velocity of each system after 5.00 s? Practice problems • The crane shown in the drawing is lifting a 180-kg crate upward with an acceleration of 1.2 m/s2. The cable from the crate passes over a solid cylindrical pulley at the top of the boom. The pulley has a mass of 130 kg. The cable is then wound onto a hollow cylindrical drum that is mounted on the deck of the crane. The mass of the drum is 150 kg, and its radius is 0.76 m. The engine applies a counterclockwise torque to the drum in order to wind up the cable. What is the magnitude of this torque? Ignore the mass of the cable.