Physics 207: Lecture 6, Pg 1 Lecture 6 Goals l Discuss uniform and non-uniform circular motion l Recognize different types of forces and know how they act on an object in a particle representation  Identify forces and draw a Free Body Diagram  Solve problems with forces in equilibrium (a=0) and non- equilibrium (a≠0) using Newton’s 1 st & 2 nd laws.

Physics 207: Lecture 6, Pg 2 Circular Motion (UCM) is common so specialized terms Angular position  (CCW + CW -) l Radius is r Arc traversed s = r  l Tangential “velocity” v T = ds/dt Angular velocity,  ≡ d  /dt (CCW + CW -) v T = ds/dt = r d  /dt = r  r  vTvT s

Physics 207: Lecture 6, Pg 3 Uniform Circular Motion (UCM) has only radial acceleration UCM changes only in the direction of v 1. Particle doesn’t speed up or slow down! 2. Velocity is always tangential, acceleration perpendicular ! 

Physics 207: Lecture 6, Pg 4 Again Centripetal Acceleration a r = v T 2 /r =  2  r Circular motion involves continuous radial acceleration vTvT r arar Uniform circular motion involves only changes in the direction of the velocity vector Acceleration is perpendicular to the trajectory at any point, acceleration is only in the radial direction.

Physics 207: Lecture 6, Pg 5 Uniform Circular Motion (UCM) is common so we have specialized terms New definitions 1. Period (T): The time required to do one full revolution, 360 ° or 2  radians 2. Frequency (f): f ≡ 1/T, number of cycles per unit time Angular velocity or speed in UCM  =  t = 2  / T = 2  f (radians per unit time) r  vTvT s

Physics 207: Lecture 6, Pg 6 Mass-based separation with a centrifuge How many g’s (1 g is ~10 m/s 2 )? a r = v T 2 / r =  2 r f = 6000 rpm = 100 rev. per second is typical with r = 0.10 m a r = (2  10 2 ) 2 x 0.10 m/s 2 BeforeAfter bb5 a r = 4 x 10 4 m/s 2 or ca g’s !!! but a neutron star surface is at m/s 2

Physics 207: Lecture 6, Pg 7 g’s with respect to humans l 1 gStanding l 1.2 gNormal elevator acceleration (up). l 1.5-2g Walking down stairs. l 2-3 gHopping down stairs. l 3.5 gMaximum acceleration in amusement park rides (design guidelines). l 4 gIndy cars in the second turn at Disney World (side and down force). l 4+ gCarrier-based aircraft launch. l 10 gThreshold for blackout during violent maneuvers in high performance aircraft. l 11 gAlan Shepard in his historic sub orbital Mercury flight experience a maximum force of 11 g. l 20 gColonel Stapp’s experiments on acceleration in rocket sleds indicated that in the g range there was the possibility of injury because of organs moving inside the body. Beyond 20 g they concluded that there was the potential for death due to internal injuries. Their experiments were limited to 20 g. l 30 gThe design maximum for sleds used to test dummies with commercial restraint and air bag systems is 30 g. Comment: In automobile accidents involving rotation severe injury or death can occur even at modest speeds

Physics 207: Lecture 6, Pg 8 A bad day at the lab…. l In 1998, a Cornell campus laboratory was seriously damaged when the rotor of an ultracentrifuge failed while in use. l Description of the Cornell Accident -- On December 16, 1998, milk samples were running in a Beckman. L2-65B ultracentrifuge using a large aluminum rotor. The rotor had been used for this procedure many times before. Approximately one hour into the operation, the rotor failed due to excessive mechanical stress caused by the g-forces of the high rotation speed. The subsequent explosion completely destroyed the centrifuge. The safety shielding in the unit did not contain all the metal fragments. The half inch thick sliding steel door on top of the unit buckled allowing fragments, including the steel rotor top, to escape. Fragments ruined a nearby refrigerator and an ultra-cold freezer in addition to making holes in the walls and ceiling. The unit itself was propelled sideways and damaged cabinets and shelving that contained over a hundred containers of chemicals. Sliding cabinet doors prevented the containers from falling to the floor and breaking. A shock wave from the accident shattered all four windows in the room. The shock wave also destroyed the control system for an incubator and shook an interior wall.

Physics 207: Lecture 6, Pg 9 Example Question l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 1 What is T the period of the initial rotation? 2 What is  the initial angular velocity? l 3 What is the tangential speed of a point on the rim during this initial period? 4 Sketch the  (angular displacement) versus time plot. l 5 What is the average angular velocity over the 1 st 10 seconds? l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the angular acceleration be? l 7 What is the magnitude and direction of the acceleration after 10 seconds?

Physics 207: Lecture 6, Pg 10 Circular Motion Radial acceleration  a r | =  v T 2 / r =  2 r Angular acceleration  = d 2  /dt =  a T | / r s = s 0 + v T0  t + ½ a T  t 2 r  vTvT s

Physics 207: Lecture 6, Pg 11 Example l A horizontally mounted disk 2.0 meters in diameter (1.0 m in radius) spins at constant angular speed such that it first undergoes (1) 10 counter clockwise revolutions in 5.0 seconds and then, again at constant angular speed, (2) 2 counter clockwise revolutions in 5.0 seconds. l 1 What is T the period of the initial rotation? T = time for 1 revolution = 5 sec / 10 rev = 0.5 s

Physics 207: Lecture 6, Pg 12 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. 2 What is  the initial angular velocity?  = d  /dt =  /  t  = 10 2π radians / 5 seconds = 12.6 rad / s ( also 2  f = 2  / T )

Physics 207: Lecture 6, Pg 13 Example l A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. l 3 What is the tangential speed of a point on the rim during this initial period? | v T | = ds/dt = (r d  /dt = r  | v T | = r  = 1 m 12.6 rad/ s = 12.6 m/s

Physics 207: Lecture 6, Pg 14 Example l A horizontally mounted disk 1 meter in radius spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds. 4 Sketch the  (angular displacement) versus time plot.  =   +    t r  vtvt s

Physics 207: Lecture 6, Pg 15 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  x 2/5) 5 rad  = 24 rad

Physics 207: Lecture 6, Pg 16 Example l 5 What is the average angular velocity over the 1 st 10 seconds?

Physics 207: Lecture 6, Pg 17 Sketch of  vs. time time (seconds)        (radians)  =   +   t  =  +  5  rad  =   +   t  =   rad + (  5) 5 rad  =  rad 5 Avg. angular velocity =  /  t = 24  /10 rad/s

Physics 207: Lecture 6, Pg 18 Then angular velocity is no longer constant so d  /dt ≠ 0 Define tangential acceleration as a T ≡ dv T /dt = r d  /dt Define angular acceleration  ≡ d  dt Let  be constant & integrating:  =  0 +   t Integrating again:  =  0 +  0  t + ½   t 2 l Multiply by r r   = r  0 + r  0  t + ½ r   t 2 s  = s 0 + v T  t + ½ a T  t 2 l Many analogies to linear motion but it isn’t one-to-one Non-uniform CM: What if  is linearly increasing …

Physics 207: Lecture 6, Pg 19 Example 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must be the angular acceleration  ? Key point …..  is associated with tangential acceleration (a T ).

Physics 207: Lecture 6, Pg 20 Tangential acceleration?  =  o +  o  t +  t 2 (from plot, after 10 seconds) r  vtvt s 1 2 aTaT r l 6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the “tangential acceleration” be? l 7 What is the magnitude and direction of the acceleration after 10 seconds? 24  rad = 0 rad + 0 rad/s  t + ½ (a T /r)  t 2 48  rad 1m / 100 s 2 = a T =0.48  m/s 2

Physics 207: Lecture 6, Pg 21 Non-uniform Circular Motion For an object moving along a curved trajectory, with varying speed Vector addition: a = a r + a T (radial and tangential) arar aTaT a

Physics 207: Lecture 6, Pg 22 Tangential acceleration? a T = 0.48  m / s 2 and v T = 0 + a T  t= 4.8  m/s = v T a r = v T 2 / r = 23  2 m/s 2 r  vTvT s l 7 What is the magnitude and direction of the acceleration after 10 seconds? Tangential acceleration is too small to plot!

Physics 207: Lecture 6, Pg 23 Chaps. 5, 6 & 7 What causes motion? (What is special about acceleration?) What are forces ? What kinds of forces are there ? How are forces and changes in motion related ?

Physics 207: Lecture 6, Pg 24 Newton’s First Law and IRFs An object subject to no external forces moves with constant velocity if viewed from an inertial reference frame (IRF). If no net force acting on an object, there is no acceleration.

Physics 207: Lecture 6, Pg 25 No Net Force, No acceleration…a demo exercise l In this demonstration we have a ball tied to a string undergoing horizontal UCM (i.e. the ball has only radial acceleration) 1 Assuming you are looking from above, draw the orbit with the tangential velocity and the radial acceleration vectors sketched out. 2 Suddenly the string brakes. 3 Now sketch the trajectory with the velocity and acceleration vectors drawn again.

Physics 207: Lecture 6, Pg 26 Lecture 6 Assignment: Read Chapter 5 &