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Physics I Class 06 Newton’s Second Law for More Complex Cases 06-1 Newton’s Second Law Yet Another Review! Newton’s Second Law: F Fnet Fnet m a or a m In a complex situation, we will need to apply this to more than one object and/or in more than one dimension. When we do this, we will get a set of linear equations and we will solve them for the unknown quantities. 06-2 Using Newton’s Second Law to Solve Complex Problems 1. 2. 3. 4. 5. 6. Identify all forces acting on the object. Today: Gravity, normal, and ropes/strings. Choose a coordinate system. If you know the direction of acceleration, one coordinate axis should be in that direction. Draw a “Free-Body Diagram.” We will use two dimensions today. Express the force vectors in components. We will use trigonometry today. Use Newton’s Second Law to write one equation for each direction considered. Solve the equations. 06-3 A Common Example: Atwood’s Machine a W = mg X W = Mg T T m w = mg a X M W = Mg 06-4 Solution to Atwood’s Machine T mg ma Mg T Ma Add these to eliminate T . Mg mg Ma ma ( M m) g ( M m)a M m a g M m 06-5 Inclined Plane 06-6 Coordinate Systems and Free-Body Diagrams Y X a X Use trigonometry to determine X & Y components of forces not aligned with coordinate system. 06-7 Solving for Acceleration For mass 1: T m1g sin( ) m1 a X: Y: N m1g cos( ) 0 For mass 2: m 2g T m 2 a X: To solve for a, add the two X equations: T m1g sin( ) m 2 g T m1 a m 2 a m m1 sin( ) a 2 g m1 m 2 What would it mean if we found a < 0 after plugging in the values? If = 0, does the “inclined” plane resemble something in class? 06-8 Class #6 Take-Away Concepts 1. 2. 3. 4. Keep using the six-step process for doing Newton’s Second Law problems for complex cases, it will help you keep things straight. For each dimension and each object, you will get one equation. You may or may not need to know the forces in the “normal” direction. You should have the same number of unknowns as equations. The easiest way to solve is usually to add the two equations resulting from opposite ends of a rope or string, or opposite sides of a contact surface where two objects push on each other. 06-9 Activity #6 - Forces and Motion in Coupled Systems Objectives of the Activity: 1. 2. Making detailed theoretical predictions and comparing with measured data. Understanding forces and motion in coupled systems. 06-10