1. Physics I LECTURE 18 11/18/09

2. Administrative Notes EXAM III, Nov. 25th
In Class, 2:30-3:20 pm
Chapters 8-10
Exam review session Tuesday (11/24) night, 6:30-9:30pm, OH 218.

3. Outline Newton’s Law of Universal Gravitation Weightlessness
Kepler’s Laws
Work by Constant Force
Scalar Product of Vectors
Work done by varying Force
Work-Energy Theorem
Conservative, non-conservative Forces
Potential Energy
Mechanical Energy
Conservation of Energy
Dissipative Forces
Gravitational Potential Revisited
Power
Momentum and Force
Conservation of Momentum
Collisions
Impulse
Conservation of Momentum and Energy
Elastic and Inelastic Collisions2D, 3D Collisions
Center of Mass and translational motion
Angular quantities
Vector nature of angular quantities
Constant angular acceleration
What do we know?
Units
Kinematic equations
Freely falling objects
Vectors
Kinematics + Vectors = Vector Kinematics
Relative motion
Projectile motion
Uniform circular motion
Newton’s Laws
Force of Gravity/Normal Force
Free Body Diagrams
Problem solving
Uniform Circular Motion

4. Review We also discussed the Center of Mass (CM)
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
Calculation of CM for any object using integration techniques

5. CM Review
Calculate the center of mass of the letter K.
45º 2
45º 1

6. Example: Triangle
Imagine we have the 2D triangle below which has a uniform density of σ=3kg/m2. 5m 2m

7. Example: Triangle
Imagine we have the 2D triangle below which has a uniform density of σ=3kg/m2. 5m 2m

8. 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.

9. 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?

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

11. 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?

12. 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!

13. 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

14. Circular Motion Nomenclature: Angular Displacement
Axis of rotation
Axis of rotation

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

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

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

18. 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? (c) and the linear and centripetal acceleration of a point on the edge of the record?

19. 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.

20. 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

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

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

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

24. 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?

25. 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?

26. Example
A centrifuge speeds up to 25,000rpm in 40s with constant angular acceleration.
A) What is the angular acceleration of the centrifuge?
B) How many revolutions has the centrifuge made by the time it gets up to speed?

27. Example
A centrifuge speeds up to 25,000rpm in 40s with constant angular acceleration.
A) What is the angular acceleration of the centrifuge?
B) How many revolutions has the centrifuge made by the time it gets up to speed?