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Kinematics in Cartesian coordinates

Kinematics in Cartesian coordinates. If. is given, you can find. and. DO NOT use these Constant acceleration formulas when acceleration is a function of time. You have to integrate or differentiate! DO NOT mix x- and y-components!. Kinematics in polar coordinates. y. . x.

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Kinematics in Cartesian coordinates

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  1. Kinematics in Cartesian coordinates If is given, you can find and

  2. DO NOT use these Constant acceleration formulas when acceleration is a function of time. You have to integrate or differentiate! DO NOT mix x- and y-components!

  3. Kinematics in polar coordinates y  x Uniform circular motion:

  4. Kinematics of circular motion If is given, you can find and by integration, similarly to linear motion

  5. Vectors Cross product, dot product

  6. Newton’s 2nd Law

  7. Newton’s 3rd Law N Ffr Ffr N

  8. Solving Problems • Sketch • Isolate the bodies, draw a free-body diagram (only external forces; use 3rd law) • 2. Write down 2nd Newton’s law for each body Choose a coordinate system Write 2nd Newton’s law in component form: 3. Solve for acceleration etc.

  9. Torque and Angular Momentum Motion in the plane, in polar coordinates: y  Kinetic energy: x

  10. Dynamics of rotational motion Central force y  O x

  11. Rigid bodies rotating about a fixed axis R m2 m1

  12. Conservation laws: • Momentum • Angular momentum • energy • These quantities are additive • P and L are vectors; only some of their components may be conserved

  13. Work Energy Theorem Central force:

  14. Mechanical energy is conserved!

  15. Know potential energy for familiar forces: • gravity near Earth, F = mg • spring force Fx = - k(x-L) Know how to find potential energy for unfamiliar forces: Fx = ax2 – bxetc. Graphic representation of motion for a given potential energy function. Identify points of equilibrium (stable and unstable) and turning points.

  16. Conservation of Momentum Sometimes only Fx or Fy may be equal to zero. Then only px or py is conserved. If F is not zero, but the collision is very short (Ft is small as compared to change in momentum), you can still use momentum conservation relating moments of time immediately before and after the collision. Only if the collision is perfectly elastic, the kinetic energy is conserved

  17. Conservation of Angular Momentum For symmetrical objects rotating about their axis of symmetry: R m2 Second Law: m1

  18. Rolling without slipping Linear velocity and acceleration of the rim with respect to the center of rotation are related to angular velocity and acceleration: Friction force is not zero but work done by this force is zero. One can apply energy conservation. Applications: rolling of rigid bodies, pulleys, yo-yos

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