1. AP Physics C Montwood High School R. Casao

2. When a wheel moves along a straight track, the center of the wheel moves forward in pure translation. A point on the rim of the wheel traces out a complex path called a cycloid.

3.  For a wheel passing at constant speed while rolling smoothly (no sliding): ◦ the center of mass O of the wheel moves forward at constant speed v com. ◦ the point P where the wheel makes contact with the surface also moves forward at speed v com so that it remains directly below the center of mass O.  During time interval t, both O and P move forward by a distance s. Observers see the wheel rotate thru an angle  about the center of the wheel. t 1 = 0t 2 = t

4.  The point on the wheel P that was touching the surface at t = 0 s moves through an arc length s.  Equations: s = R·  ; v com = ds/dt;  = d  /dt  Differentiating the arc length equation wrt time: ds/dt = R·d  /dt to give us v com = R· . t 1 = 0t 2 = t

5.  The rolling motion of a wheel is a combination of purely translational and purely rotational motions.  Every point on the wheel rotates about the center with angular speed .  Every point on the outside edge of the wheel has linear speed v com.  Purely translational motion: every point on the wheel moves forward with speed v com.  Combination of rotational motion and translational motion produces the rolling motion of the wheel.

7.  The portion of the wheel at the bottom (point P) is stationary and the top of the wheel is moving at speed 2·v com. The wheel is moving fastest near the top than near the bottom because the spokes are more blurred at the top than at the bottom.

8.  Consider the rolling motion about an axis thru point P: K = 0.5·I p ·  ² ◦ I p is the rotational inertia of wheel about axis thru point P and  is the angular speed of the wheel. ◦ Apply the parallel-axis theorem to determine I p : I p = I com + M·h²; for the wheel, h = R  Substituting: K = 0.5·I com ·  ² + 0.5·M·R²·  ²  Because v com = R· , K = 0.5·I com ·  ² + 0.5·M·v com ²

9.  A rolling object has two types of kinetic energy: a rotational kinetic energy (0.5·I com ·  ²) due to its rotation about its center of mass and a translational kinetic energy (0.5·M·v com ²) due to translation of its center of mass. Forces of Rolling  If a wheel rolls at constant speed it has no tendency to slide at the point of contact P and no frictional force acts there; a com = 0 at point P.  If a net force acts on the rolling wheel to speed it up or slow it down, the net force causes an acceleration of the center of mass along the direction of travel.

10.  The wheel also rotates faster or slower, so an angular acceleration  occurs about the center of mass.  These accelerations tend to make the wheel slide at point P and a frictional force must act on the wheel at P to oppose the tendency to slide.  If the wheel does NOT slide, the force is a static frictional force f s and the motion is smooth rolling motion.  Smooth rolling motion: a com = R·   If the wheel does slide when the net force acts on it, the frictional force that acts at P is a kinetic frictional force and the motion is not smooth rolling motion.

11.  For a uniform body of mass M and radius R rolling smoothly down a ramp at angle  along an x axis.  The normal force N acts at point P but has been shifted up to the center of mass.  Static frictional force f s acts at point P and is directed up the incline. If the body were to slide down the incline, the frictional force would oppose that motion. a com

12.  Motion along the x-axis (up as positive): f s – M·g·sin  = M·a com  Angular motion: Torque = I·  ; ◦ N and F g ·sin  pass thru the pivot at the center of mass and do not produce a torque. ◦ f s applied perpendicularly a distance R at point P and produces a torque. ◦ Torque = F  ·r  = f s ·R; f s ·R = I com ·  ◦ -a com = R·  ; a com is negative because it is directed in the negative direction on the x-axis. Unlike the linear motion problems we have done in the past where we took the direction of motion as positive, with the rotation involved, we take the positive direction as the positive direction of the rotation (ccw = positive; cw = negative).

13.  Solve for  and substitute:  =-a com /R f s ·R = I com · -a com /R

14.  As yo-yo rolls down its string from height h, potential energy m·g·h is converted to both rotational (0.5·I com ·  ²) and translational kinetic energy (0.5·M·v com ²). As it climbs back up, kinetic energy is converted to potential energy.  The string is looped around the axle and when the yo-yo hits the bottom of the string, an upward force on the axle from the string stops the descent.  The yo-yo spins with the axle inside the loop with only rotational kinetic energy. y a com

15.  The yo-yo spins until you jerk on the string to cause the string to catch on the axle and allow the yo-yo to climb back up.  The rotational kinetic energy of the yo-yo at the bottom of the string can be increased by throwing the yo-yo downward so that it starts down the string with initial speeds v com and  instead of rolling down from rest.  The yo-yo rolls on the axle of radius R o.  The yo-yo is slowed by the tension force T from the string on the axle. y a com

16.  Acceleration equation:  the yo-yo has the same downward acceleration when it is climbing back up the string because the forces acting on it are still those shown in the figure. Torque Revisited  We previously defined torque for a rigid body that rotated about a fixed axis in a circle.  To define the torque for an individual particle that moves along any path relative to a fixed point (rather than a fixed axis), the path doesn’t have to be a circle and the torque has to be written as a vector.

17.  The figure shows a particle at point A in the xy plane with a single force F acting on the particle.  The particle’s position relative to the origin O is given by the position vector r.  Torque equation:

18.  To find the direction of the torque, slide the force vector without changing its direction until its tail is at the origin O so that it is tail to tail with the position vector.  Use the right hand rule to rotate the position vector r into the force vector F. The thumb points in the direction of the torque.

19.  The magnitude of the torque is given by:   is the angle between the position vector r and the force vector F. Angular Momentum  The figure shows a particle of mass m with linear momentum p = m·v as it passes thru point a in the xy plane. The angular momentum l of the particle with respect to the origin O is:

20.  r is the position vector of the particle wrt O.  As the particle moves wrt O in the direction of its momentum p, the position vector r rotates around O.  To have angular momentum, the particle itself does not have to rotate around O.  Unit for angular momentum: kg·m²/s = J·s  The direction of the angular momentum vector is found by sliding the vector p until its tail is at the origin O.

21.  Use the right-hand rule, rotating the vector r into vector p. The thumb points in the direction of the angular momentum vector.  The magnitude of the angular momentum vector is: l = r·m·v·sin , where  is the angle between r and p when the two vectors are tail to tail.  Angular momentum only has meaning wrt a specific origin.  The direction of the angular momentum vector is always perpendicular to the plane formed by the position vector and the linear momentum vector.

22.  F net = dp/dt expresses the close relation between force and linear momentum for a single particle.  There is also a close relationship between torque and angular momentum: T net = dl/dt  The vector sums of all the torques acting on a particle is equal to the time rate of change in the angular momentum of that particle.  The torques and angular momentum are both defined wrt the same origin.

23.  For a system of particles with respect to an origin, the total angular momentum L of the system is the vector sum of the angular momenta l of the individual particles:  Over time, the angular momenta of individual particles may change due to interactions within the system between individual particles or because of outside influences on the system.

24.  Since T net = dl/dt, then  The rate of change of the system’s angular momentum L is equal to the vector sum of the torques on the individual particles.  These torques include internal torques due to forces between particles and external torques due to forces on the particles from bodies outside of the system.  The forces between particles always occur in 3 rd law pairs so their torques cancel each other, so the only torques that can change the total angular momentum L of a system are external torques.

25.  2 nd law for rotation of a system of particles: the net external torque T net acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L.  The torque and the system’s angular momentum must be measured from the same origin. Angular Momentum of a Rigid Body Rotating About a Fixed Axis  For a system of particles that form a rigid body that rotates about a fixed axis rotating with constant angular speed  : L = I· 

26.  If the net external torque acting on a system is zero, the angular momentum L of the system remains constant, no matter what changes take place within the system: L i = L f ; I i ·w i = I f ·w f  Because momentum is a vector quantity, the conservation of momentum has to be considered in all three dimensions (x, y, z). Depending on the torques acting on a system, the angular momentum of the system might be conserved in only 1 or 2 directions but not in all directions.

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