Wednesday 6/10PHYS 2010

Nathalie HoffmannUniversity of Utah

Torque

• 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

• 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

• I is moment of inertia• Ω is angular velocity

• 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.