Chapter 7 Extra Topics Crater Lake Oregon Photo

Centers of Mass: Torque is a WIfunction of force and distance. Lake Superior, Washburn,

If the forces are all gravitational, then If the net torque is zero, then

If we divide Mo by the total mass, we can find the center of

For a thin rod or strip: d = density per unit length (d is

For a two dimensional shape, we need two distances to locate the center of

Note: The centroid does not have to be on the object. If the center

Theorems of Pappus: When a two dimensional shape is rotated about an axis: Volume

We can find the centroid of a semi-circular surface by using the Theorems of

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

Centers of Mass: Torque is a WIfunction of force and distance. Lake Superior, Washburn, Photo by Vickie Kelly, 2004 (Torque is the tendency of a system to rotate about a point. )

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.

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

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.

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

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

coordinate of centroid = (2. 25, 2. 7)

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

Theorems of Pappus: When a two dimensional shape is rotated about an axis: Volume = area. distance traveled by the centroid. Surface Area = perimeter. distance traveled by the centroid of the arc. Consider an 8 cm diameter donut with a 3 cm diameter cross section: 1. 5 2. 5

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