1 Honors Physics 1 Class 15 Fall 2013 Rolling motion Noninertial reference frames Fictitious forces

2 Rolling motion We will treat the case where 1)the axis of rotation passes through the center of mass. 2)the axis of rotation does not change direction. Example = Rolling wheel

Rolling

The Kinetic Energy of Rolling If we view the rolling as pure rotation about an axis through P, then (  is the angular speed of the wheel and I P is the rotational inertia of the wheel about the axis through P). Using the parallel-axis theorem (I P = I com +Mh 2 ), (M is the mass of the wheel, I com is its rotational inertia about an axis through its center of mass, and R is the wheel’s radius, at a perpendicular distance h). Using the relation v com =  R, we get: A rolling object, therefore has two types of kinetic energy: 1.a rotational kinetic energy due to its rotation about its center of mass (=½ I com  2 ), 2.a translational kinetic energy due to translation of its center of mass (=½ Mv 2 com )

5 The Forces of Rolling: Friction and Rolling A wheel rolls horizontally without sliding while accelerating with linear acceleration a com. A static frictional force f s acts on the wheel at P, opposing its tendency to slide. The magnitudes of the linear acceleration a com, and the angular acceleration  can be related by: where R is the radius of the wheel.

6 The Forces of Rolling: Rolling Down a Ramp A round uniform body of radius R rolls down a ramp. The forces that act on it are the gravitational force F g, a normal force F N, and a frictional force f s pointing up the ramp.

7 Sample problem: Rolling Down a Ramp A uniform ball, of mass M=6.00 kg and radius R, rolls smoothly from rest down a ramp at angle  =30.0°. a) The ball descends a vertical height h=1.20 m to reach the bottom of the ramp. What is its speed at the bottom? Calculations: Where I com is the ball’s rotational inertia about an axis through its center of mass, v com is the requested speed at the bottom, and  is the angular speed there. Substituting v com /R for , and 2/5 MR 2 for I com, b) What are the magnitude and direction of the frictional force on the ball as it rolls down the ramp? Calculations: First we need to determine the ball’s acceleration a com,x : We can now solve for the frictional force:

8 A wheel and a cube, both of mass M are made to race down a slope. The wheel rolls without slipping. The cube slides without rolling. Which one gets to the bottom first? Which one has greater KE at the bottom?

9 Energy analysis of rolling down a ramp a)A cubical block with mass M and side 2R is allowed to slide without rolling down a ramp/slide from height h to height 0. How fast does the center of mass travel at the bottom of the slope? b)A wheel with mass M and radius R is allowed to roll without slipping on a ramp of height h to the level ground at height 0. How fast does its center of mass travel at the bottom of the ramp?

10 The Yo-Yo

11 Galilean transformations Consider measurements of the motion of a mass m made from each of two coordinate systems. For simplicity let’s assume that corresponding axes are parallel and the scale units (time and distance) are the same. The origins of the two systems are displaced by S so

12 Galilean transformations 2

13 XKCD LINK

14 Linearly Accelerating Frame Fictitious forces How is Newton’s second law affected by measurement in accelerating frames?

15 Example: Hanging mass in a car T T fict mg

16 Principle of Equivalence In the example problem, we treated acceleration A in the same way as we treated gravitational acceleration. The Principle of Equivalence states that there is no way to distinguish locally* between a gravitational acceleration and an acceleration of the coordinate system. *Locally means that we don’t look outside the system for the cause of an acceleration or gravity.

17 Rotating motion and fictitious forces Consider two reference frames, one rotating about the z axis, the other inertial. Origins coincide. Z axes coincide. What fictitious forces are observed in the rotating frame? We already have some experience with this from analyzing forces in the spinning terror ride and centripetal acceleration.

18 Rotating frames

19 Rotating frames

20 Fictitious forces in rotating frames