# June 10 ```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.
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