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Physics 106 Lesson #2 Static Equilibrium Dr. Andrew Tomasch 2405 Randall Lab [email protected] Last Time: Scalars & Vectors • Scalars: magnitude only (ex: radius R =10 cm) • Vectors: magnitude and direction – Arrow length ≡ magnitude of the vector – Arrow direction ≡ vector direction Example: d Displacement ≡ 1m d d = 1 m to the right magnitude direction The Net (Total) Force • Because the washer is at rest the net (total) force on it is zero • The net force is the vector sum of all the forces acting on the object • How do we add vectors? Spring Force Fs= kx ? Washer Weight W = mg Vector Addition Caution Quiz Ahead A B ? • Place vectors “tip to tail” and draw an arrow from the start (tail) of one vector to the end (tip) of the other B • How about A-B ? A+B -B A A-B A A-B = A+(-B) B Caution Quiz Ahead Concept Test #1 Two vectors, one with magnitude 3 m and the other with magnitude 4m, are added together. The resultant vector could have a magnitude as small as: 1 m 4m A) 1m B) 3m C) 5m D) 7m 3m The resultant has the largest magnitude when the two vectors are parallel and the smallest when they are antiparallel. Concept Test #2 Two vectors have unequal magnitudes. Can their sum be zero? 0 • Yes • No The smallest possible vector sum Is when the two vectors are antiparallel. The only way for this sum to be zero is If the vectors have equal magnitudes. Equilibrium Condition for a Body at Rest • For a body to be in equilibrium and remain at rest: • An example: F 0 Spring Force F 0 Fs= kx Shorthand for the net force: “The vector sum of all forces” Weight W = mg Washer + =0 Torque: t A force which points through the rotation axis produces no torque because r = 0! • Torque t (a vector): Magnitude ≡ the product of a perpendicular force and the distance r to a rotation axis: t t rF t "The magnitude (length) of vector t " F is perpendicular to r r F The distance r is also called the lever arm Torque can also be defined more generally for forces that are not perpendicular to r by expressing F as the sum of two component vectors, one parallel to the door and one perpendicular to it. Only the perpendicular component vector produces a torque about the hinge to make the door rotate. F F t F r t Torque Continued • Direction (looking down the rotation axis): – Counterclockwise rotation is “(+)” (positive) – Clockwise rotation is “(-)” (negative) – Direction in space by Right Hand Rule (RHR) • Why a perpendicular force F ? – Because only a perpendicular force can produce a rotation • Units: N m t rF Caution Quiz Ahead points t out of page F Arrow out of page Arrow into page Concept Test #3 Both r and F remain unchanged. F does not change magnitude or direction → torque remains unchanged. You are trying to open a door that is stuck by pulling on the knob in a direction perpendicular to the door. If you instead tie a rope to the knob and pull with the same magnitude force in the same direction, does the torque you exert increase? • • Yes No r F Concept Test #4 You are using a wrench to try to tighten a nut. Which of the arrangements shown is most effective for tightening the nut? List in order of decreasing effectiveness. (The “rod” can be used to extend the wrench handle length). The force F is the same in each case. F F r rod F F r r rod r 3 1 2 4 The longest lever arm r exerts the largest torque for the same applied force A. 2 > 1 > 3 > 4 B. 2 > 1 = 4 > 3 C. 4 > 2 > 1 > 3 D. 2 > 3 > 4 > 1 Equilibrium for Rigid Bodies • Extended objects which do not change shape are called rigid bodies • For a rigid body to be in equilibrium: F 0 t 0 • For a rigid body to remain stationary in one place, the net force acting on it must be zero • For a rigid body to remain stationary and not rotate the net torque about any axis through the body must be zero • The net force/torque is the vector sum of all forces/torques acting on the body (analyze with Free Body Diagram) Center of Mass/Gravity • • • • • There is a special geometrical point associated with any rigid body (extended object) where we can attribute the body’s mass and weight as acting at that point This special point is called the center of mass (CM) and the center of gravity (CG) of the object The definitions for center of mass and center of gravity are different, but if the acceleration of gravity g is constant over the extent of the object, then the center of mass and the center of gravity are located at the same geometrical point. For uniform objects this will be the geometrical center of the object The motion of any rigid body of mass m can be described as the trajectory through space of a point mass m located at its center of mass on which is superposed rotation of the object about the center of mass For the purpose of calculating torques, all the weight of the extended object can be assumed to act at the center of gravity Demo: Dumbbell With Lights Where Is the Center of Gravity of This Yummy Donut? Center of Gravity (CG) The Center of gravity does not need to be located inside the object. Sometimes the CG can be located in a surprising place… It is at the center of the circular ring, half way from the bottom of the donut - where there is no dough! After A Bite… Center of Gravity If it falls within the object, the center of gravity corresponds to the point where an object balances. Wine Bottle Balance • Treat the wine rack and the bottle as a rigid body • Two forces: – Supporting force FN from table – Weight at the center of gravity of the system • Net torque = 0 →No rotation → • The center of gravity is located exactly above the point where the supporting force acts • The supporting force FN from the table is also called the “Normal” (perpendicular) force CG W FN Axis of rotation Neither force produces a torque about the Axis of Rotation since both forces point through the axis so that r = 0 for both forces Wine Bottle Balance • The wine bottle and seesaw are balanced in the same way! FN CG Axis of rotation W CG W FN Axis of rotation