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Advanced Force Analysis in Two Dimensions

Explore equilibrium, accelerated motion, and handling forces in multiple objects with 2-D Newton’s 2nd Law. Learn to solve complex force problems using x-y components. Practice with sample scenarios involving inclined planes and frictional forces.

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Advanced Force Analysis in Two Dimensions

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  1. Chapter 4 Revisited • Forces in two dimensions • Analyzing force problems is only slightly more complicated when forces act in more than one dimension.

  2. Newton’s 2nd Law in 2-D • You must still identify all forces and draw your free body diagram. • Choose a coordinate system. • You then resolve your problem into an x-problem and a y-problem (like projectile motion).

  3. Equilibrium • An object either at rest or moving with a constant velocity is said to be in equilibrium • The net force acting on the object is zero so • In two dimensions this becomes,

  4. Accelerated motion • Similar to equilibrium except • Use components

  5. Multiple objects • If more than one object is present, draw free body diagram for each object • Apply equations, keeping track of signs

  6. Sample problem Larry pushes a 200 kg block on a frictionless floor at a 45o angle below the horizontal with a force of 150 N while Moe pulls the same block horizontally with a force of 120 N. a) Draw a free body diagram. b) What is the acceleration of the block? c) What is the normal force exerted on the block?

  7. Inclined Planes • Choose the coordinate system with x along the incline and y perpendicular to the incline • Replace the force of gravity with its components • The normal force is perpendicular to inclined surfaces. It’s always equal to the component of weight perpendicular to the surface. x

  8. Sample problem • Find the normal force exerted on a 2.5-kg book resting on a surface inclined at 28above the horizontal. • If the angle of the incline is reduced, do you expect the normal force to increase, decrease, or stay the same?

  9. Sample Problem How long will it take a 1.0 kg block initially at rest to slide down a frictionless 20.0 m long ramp that is at a 15o angle with the horizontal?

  10. fs Static friction on a ramp surface N Without friction, the book will slide down the ramp. If it stays in place, there is sufficient static friction holding it there. Physics W = mg q Wx = mgsinq and N = mgcosq At maximum angle before the book slides, we can prove that ms = tanq.

  11. fs Wx q Static friction on a ramp x surface N Assume q is maximum angle for which book stays in place. SF = 0 Wx = fs mgsinq = msmgcosq ms = sinq/cosq = tanq Physics W = mg q fs = mgsinq and N = mgcosq At maximum angle before the book slides, we can prove that ms = tanq.

  12. Problem A 900kg polar bear slides down a snow bank at an angle of 25 degrees. The coefficient of friction between the bear and the snow is .05; find the force of friction.

  13. Problem A 10-kg wooden box slides on a wooden ramp. The coefficient of kinetic friction is 0.30. • What is the friction force between the box and ramp if the ramp is at a 45oangle? • What is the acceleration of the box?

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