center of mass and stable equilibrium

• Find center of mass of shapes – Balance on finger, push on table, • Find center of mass of T • Find center of mass of ball bat, hockey stick • Human balancing • Balls of clay on sticks – Stable equilibrium – Neutral equilibrium – Unstable equilibrium – Balancing toys

© 2014 Pearson Education, Inc. pg # Center of mass

• An object will freely hang so that it’s center of mass is directly below the pivot. • An object will balance when the COM is above the base. • Center of mass (COM) of an object is a point where a force exerted on the object pointing directly toward or away from that point will not cause the object to turn.

© 2014 Pearson Education, Inc. pg # Center of mass One point that moves in same path a particle would if subjected to same net force.

© 2014 Pearson Education, Inc. pg # Tip • COM depends on mass distribution, but the mass of the object is not necessarily evenly distributed around the COM. • Don’t take the name of this point literally.

© 2014 Pearson Education, Inc. pg # Equilibrium and tipping objects • It is easier to balance while standing in a moving bus or train if you spread your feet apart in direction of motion. • You increase area of support or base—area of contact between an object and surface it is supported by.

© 2014 Pearson Education, Inc. pg # Figure 9-17 Humans adjust their posture to achieve stability when carrying loads.

© 2014 Pearson Education, Inc. pg # Equilibrium and COM Stable equilibrium – rotation will raise the COM Unstable equilibrium – rotation will lower the COM Neutral equilibrium – rotation will neither raise of lower COM

© 2014 Pearson Education, Inc. pg # Equilibrium and tipping objects • If a vertical line through its center of mass passes through area of support object won’t tip. • If area of support is large or if COM is closer to ground, more tipping is possible without object falling over; it is more stable.

© 2014 Pearson Education, Inc. pg # Where is the gravitational force exerted on a rigid body? • If the object does not tip, all forces exerted on it pass through COM. • We assume gravitational force exerted on object is exerted at COM. • COM is sometimes called center of gravity. • COM and center of gravity are same unless an object is so large that the gravitational field strength (g) changes for different parts i.e. Mt Everest.

© 2014 Pearson Education, Inc. pg # Mass distribution and center of mass • When an object is in static equilibrium, torques produced by weight on each side of COM have equal magnitudes.

© 2014 Pearson Education, Inc. pg # Center of mass: Quantitative definition

• Using these equations for an object with a continuous mass distribution involves calculus.

© 2014 Pearson Education, Inc. pg # Mass distribution and center of mass

To find the center of mass, we need a coordinate system with an origin.

© 2014 Pearson Education, Inc. pg #