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Physics I 95.141 LECTURE 18 11/15/10

Physics I 95.141 LECTURE 18 11/15/10. Outline/Notes . Outline Center of Mass Angular quantities Vector nature of angular quantities Constant angular acceleration. Administrative Notes HW review session moved to Thursday, 11/18, 6:30pm in OH218. Review.

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Physics I 95.141 LECTURE 18 11/15/10

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  1. Physics I95.141LECTURE 1811/15/10

  2. Outline/Notes Outline Center of Mass Angular quantities Vector nature of angular quantities Constant angular acceleration Administrative Notes HW review session moved to Thursday, 11/18, 6:30pm in OH218.

  3. Review • In the previous lecture we discussed collisions in 2D and 3D. • Momentum always conserved! Can write a conservation of momentum expression for each dimension/component. • If the collision is elastic, then we can also say that Kinetic Energy is conserved, and include this in our equations: • We also discussed the Center of Mass (CM) • Calculation of CM for 1D point masses • Calculation of CM for 3D point masses • Calculation of CM for symmetric solid objects

  4. Solid Objects • We can easily find the CM for a collection of point masses, but most everyday items aren’t made up of 2 or 3 point masses. What about solid objects? • Imagine a solid object made out of an infinite number of point masses. The easiest trick we can use is that of symmetry!

  5. Solid Objects (General) • If symmetry doesn’t work, we can solve for CM mathematically. • Divide mass into smaller sections dm.

  6. Solid Objects (General) • If symmetry doesn’t work, we can solve for CM mathematically. • Divide mass into smaller sections dm.

  7. Example: Rod of varying density • Imagine we have a circular rod (r=0.1m) with a mass density given by ρ=2x kg/m3. x L=2m

  8. Example: Rod of varying density • Imagine we have a circular rod (r=0.1m) with a mass density given by ρ=2x kg/m3. x L=2m

  9. CM and Translational Motion • The translational motion of the CM of an object is directly related to the net Force acting on the object. • The sum of all the Forces acting on the system is equal to the total mass of the system times the acceleration of its center of mass. • The center of mass of a system of particles (or objects) with total mass M moves like a single particle of mass M acted upon by the same net external force.

  10. Example • A 60kg person stands on the right most edge of a uniform board of mass 30kg and length 6m, lying on a frictionless surface. She then walks to the other end of the board. How far does the board move?

  11. CM Review • What is the center of mass of the shape below, if we assume a constant surface density (σ [kg/m2])? 1m 1m 6m 4m 1m 4m (0,0)

  12. CM Review • Calculate motion of the letter K (total mass MK=2kg) if a Force is applied to the letter.

  13. Motion of an object/system under a Force • We know that for a system of masses, or for a solid object, if a Force is applied to the system/object, the center of mass of the moves as if all of the mass was at the CM and the Force is applied to the CM. • But does this entirely determine the motion of the object?

  14. Rotation • Objects don’t only move translationally, but can also vibrate or rotate. • In this chapter (10) we are going to look at rotational motion. • First, we need to go back and review the nomenclature we use to describe rotational motion. • Motion of an object can be described by translational motion of the CM + rotation of the object around its CM!

  15. Circular Motion Nomenclature: Angular Position • It is easiest to describe circular motion in polar coordinates. y For θ in radians!!! R R x Axis of rotation

  16. Circular Motion Nomenclature: Angular Displacement • Angular displacement R Axis of rotation Axis of rotation

  17. Circular Motion Nomenclature: Angular Velocity and Acceleration • Average Angular Velocity • Instantaneous Angular Velocity • Average Angular acceleration • Instantaneous Angular acceleration

  18. Usefulness of Angular Quantities • Each point on a rotating rigid body has the same angular displacement, velocity, and acceleration! • What about translational quantities?

  19. Tangential Acceleration • If we can calculate tangential velocity from angular velocity and radius: • We can also calculate tangential acceleration: • So, total acceleration is:

  20. Example • A record (r=15cm), starting from rest, accelerates with a constant angular acceleration α=0.2 rad/s for 5 seconds. What is (a) the angular velocity of the record at t=5s? (b) the linear velocity of a point on the edge of the record (t=5s)? (c) and the linear and centripetal acceleration of a point on the edge of the record (t=5s)?

  21. Frequency and Period • We can relate the angular velocity of rotation to the frequency of rotation: • Can also write the period in terms of angular velocity, but Period (T) only makes sense for uniform circular motion.

  22. Vector Nature of Angular Quantities • We can treat both ω and α as vectors • If we look at points on the wheel, they all have different velocities in the xy plane • Choosing a vector in the xy plane doesn’t make sense • Choose vector in direction of axis of rotation • But which direction? z • Right Hand Rule • Use fingers on right hand to trace rotation of object • Direction thumb points is vector direction for angular velocity, acceleration

  23. Constant Angular Acceleration • In Chapter 2, we discussed the kinematic equations for motion with constant acceleration.

  24. Rotational vs. Translational Equations of Motion • The equations of motion for translational motion and rotational motion are parallel! • Makes it very easy to remember!

  25. Constant Angular Acceleration • If you can remember your kinematic equations for translational motion, you can solve problems with constant angular acceleration!

  26. Example • A top is brought up to speed with α=7rad/s2 in 1.5s. After that it slows down slowly with α=-0.1rad/s2 until it stops spinning. • A) What is the fastest angular velocity of the top? • B) How long does it take the top to stop spinning once it reaches its top angular velocity? • C) How many rotations does the top make in this time?

  27. Example • A top is brought up to speed with α=7rad/s2 in 1.5s. After that it slows down slowly with α=-0.1rad/s2 until it stops spinning. • C) How many rotations does the top make in this time?

  28. What Did We Learn Today? • Center of Mass • Symmetry • Integration • Translational Motion of… • Angular Motion • Nomenclature for angular motion • Angular displacement • Angular velocity • Angular acceleration • Constant angular acceleration • Symmetry with equations of translational motion

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