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Understanding Torque and Centers of Mass: Concepts from Physics

This chapter discusses the principles of torque and the concept of the center of mass, crucial in understanding the balance and rotation of systems. Torque is defined as the tendency of a system to rotate about a point, with net torque determining balance. For gravitational forces, calculations simplify, revealing the center of mass based on density and mass distribution. The text delves into methods for finding centers of mass for various shapes and theorems relevant to two-dimensional objects, improving comprehension of physical systems in equilibrium.

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Understanding Torque and Centers of Mass: Concepts from Physics

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  1. Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington Chapter 7 Extra Topics Crater Lake, Oregon

  2. Centers of Mass: Torque is a function of force and distance. (Torque is the tendency of a system to rotate about a point.)

  3. If the forces are all gravitational, then If the net torque is zero, then the system will balance. Since gravity is the same throughout the system, we could factor g out of the equation. This is called the moment about the origin.

  4. If we divide Moby the total mass, we can find the center of mass (balance point.)

  5. For a thin rod or strip: d = density per unit length (d is the Greek letter delta.) moment about origin: mass: center of mass: For a rod of uniform density and thickness, the center of mass is in the middle.

  6. strip of massdm distance from the y axis to the center of the strip distance from the x axis to the center of the strip Moment about x-axis: Center of mass: Moment about y-axis: Mass: For a two dimensional shape, we need two distances to locate the center of mass. y x x tilde (pronounced ecks tilda)

  7. For a plate of uniform thickness and density, the density drops out of the equation when finding the center of mass. For a two dimensional shape, we need two distances to locate the center of mass. y x Vocabulary: center of mass = center of gravity = centroid constant density d = homogeneous = uniform

  8. coordinate of centroid = (2.25, 2.7)

  9. square rectangle circle right triangle Note: The centroid does not have to be on the object. If the center of mass is obvious, use a shortcut:

  10. When a two dimensional shape is rotated about an axis: Consider an 8 cm diameter donut with a 3 cm diameter cross section: 1.5 2.5 Theorems of Pappus: Volume = area . distance traveled by the centroid. Surface Area = perimeter . distance traveled by the centroid of the arc.

  11. We can find the centroid of a semi-circular surface by using the Theorems of Pappus and working back to get the centroid. p

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