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Two-Dimensional Forces in Equilibrium

This chapter discusses the concept of equilibrium in two-dimensional forces, including inclined planes, friction, and applications involving friction and inclines. It also explores the Atwood machine and static equilibrium.

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Two-Dimensional Forces in Equilibrium

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  1. Chapter 5 Two Dimensional Forces

  2. 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 (since the acceleration is zero)

  3. Equilibrium cont. • Easier to work with the equation in terms of its components:

  4. 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

  5. Forces of Friction • When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion • This is due to the interactions between the object and its environment • This is resistance is called friction

  6. More About Friction • Friction is proportional to the normal force • The force of static friction is generally greater than the force of kinetic friction • The coefficient of friction (µ) depends on the surfaces in contact • The direction of the frictional force is opposite the direction of motion • The coefficients of friction are nearly independent of the area of contact

  7. Applications Involving Friction, Inclines On a microscopic scale, most surfaces are rough. The exact details are not yet known, but the force can be modeled in a simple way. For kinetic – sliding – friction, we write: is the coefficient of kinetic friction, and is different for every pair of surfaces.

  8. Kinetic Friction, ƒk • The force of kinetic friction acts when the object is in motion • ƒk = µ n • Variations of the coefficient with speed will be ignored

  9. 4-8 Applications Involving Friction, Inclines

  10. Static Friction, ƒs • Static friction acts to keep the object from moving • If F increases, so does ƒs • If F decreases, so does ƒs • ƒs µ n

  11. Applications Involving Friction, Inclines The static frictional force increases as the applied force increases, until it reaches its maximum. Then the object starts to move, and the kinetic frictional force takes over.

  12. Block on a Ramp, Example • Axes are rotated as usual on an incline • The direction of impending motion would be down the plane • Friction acts up the plane • Opposes the motion • Apply Newton’s Laws and solve equations

  13. Atwood Machine

  14. Atwood Machine • Let’s take a look at the forces on each mass…

  15. Atwood Machine

  16. Atwood Machine

  17. Static Equilibrium • Condition of an object when net forces equal zero • Object is motionless

  18. Hanging sign f.b.d. T2 T1 1 2 mg Free Body Diagram Since the sign is not accelerating in any direction, it’s in equilibrium. Since it’s not moving either, we call it Static Equilibrium. Thus, red + green + black = 0.

  19. T2 T1 T2 sin 2 (Y component) T1 sin 1 (y component) 1 2 T1 cos 1 (x component) T2 cos 2 (X component) Horizontal: T1 cos 1 = T2 cos 2 mg Vertical: T1 sin 1 + T2 sin 2 = mg Components & Scalar Equations If in Equilibrium……..the following would be true

  20. Sample Problem A mother and daughter are outside playing on the swings. The mother pulls the daughter and swing (total mass 55.0 kg) back so that the swing makes an angle of 40.0° with the vertical (50.0 ° from horizontal) What is the tension in each chain holding the swing seat and the daughter? A. 703N

  21. 160N 150N 40 ° 45° 75N Sample Problem… • Is this a case of equilibrium? • Calculate the magnitude of the net force

  22. Equilibrium Example – Free Body Diagrams

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