Wed. March 9th 1

PHSX213 class• Class stuff

– HW6W returned.

– HW7. Worth 1.5 times normal assignment. Due Wed Mar. 16th 6PM.

– Practice Exam.

– Projects …

– MidTerm2 : Wed. March 16th 8:00 – 9:30 PM. Same place as before (Budig 120).

– No classes next Wed. and Fri. (as planned)

• More ROTATION

Wed. March 9th 2

Rotational Inertia Demo

Wed. March 9th 3

Work and Rotational KE

• We saw for linear motion that,W = ∫ F dx = K = Kf – Ki = ½ m (vf

2 – vi2)

For rotational motion about a fixed axis,

W = ∫ d = K = Kf – Ki = ½ I (f2 – i

2)

Wed. March 9th 4

Example 10.78• Pulley, uniform disk, mass

M=0.5 kg, radius R=0.12m.

• Mass m1=0.4 kg, m2 = 0.6 kg

• Cord doesn’t slip. Disk rotates freely (ignore friction).

• What is the magnitude of the acceleration of the blocks?

• What is the tension in each cord ?

Wed. March 9th 5

Reading Quiz

• Which force gives the torque needed for objects to roll without slipping:

• A) normal force

• B) force due to gravity

• C) kinetic friction

• D) static friction

do rolling example demo in parallel

Wed. March 9th 6

Reading Quiz

• Which force gives the torque needed for objects to roll without slipping:

• A) normal force

• B) force due to gravity

• C) kinetic friction

• D) static friction

Wed. March 9th 7

Check-Point 1

• A wheel rolls without slipping along a horizontal surface. The center of the disk has a translational speed, v. The uppermost point on the wheel has a translational speed of :

• A) 0• B) v• C) 2v• D) need more information

Wed. March 9th 8

Rolling

A rolling object can be considered to be rotating about an axis through the CoM while the CoM moves.

Wed. March 9th 9

Rolling

• The rotational motion obeys = ICoM while the translational motion obeys F = m aCoM

• Note that point P in contact with the ground has zero velocity ie. stationary. So static friction is what matters at that point. – Not convinced it is stationary – think about clean (unsmudged) tire

tracks in the snow

Wed. March 9th 10

Check-Point 2

A solid disk and a ring roll down an incline. The ring is slower than the disk if

A. mring= mdisk, where m is the inertial mass.B. rring = rdisk, where r is the radius.C. mring = mdisk and rring = rdisk.D. The ring is always slower regardless of the relative values of m and r.

Wed. March 9th 11

Forces in Rolling

• Rolls smoothly down the ramp without sliding/slipping.

• How much torque about the CoM does each force produce ?

• How big is fS ?• How big is the

acceleration ?Note : I’ll define +x down the incline

Wed. March 9th 12

Rolling

• One can derive that :

• fS = I a/R2 = [/(1+)] mg sin

• And that, • a = g sin/(1 + I/(MR2)) = g sin/(1+)

• Where we have defined , via I = M R2

• hoop = 1, cylinder = 0.5, solid sphere = 0.4

Wed. March 9th 13

Discuss rolling in terms of KE

• From energy conservation considerations.• Note the frictional force in this case doesn’t

oppose the angular motion, and the work done by this non-conservative force is transformed into rotational kinetic energy.– W = ∫ d

• And we find that K = ½ m v2 + ½ I 2

• = ½ m (1+) v2 ( = mg h)

Since the contact point doesn’t move, there is no translational displacement.

Wed. March 9th 14

Another way to look at this

• Stationary observer sees rotation about an axis at point P with = v/R.

• Using the parallel axis theorem, IP = ICoM + mR2

• So, K = ½ IP 2

• = ½ ICoM 2 + ½ m v2

• just as before.

Wed. March 9th 15

Rolling Body on Inclined Plane Demo

Wed. March 9th 16

Loop-the-loop

Wed. March 9th 17

Torque Definition for a Particle

= r F

NB Only makes sense to talk about the torque wrt or about a certain point

Vector product means that the torque is directed perpendicular to the plane formed by (r, F). Whether it is up or down is from the right-hand rule convention

Wed. March 9th 18

Angular Momentum Demo

Wed. March 9th 19

Angular Momentum Definition for a Particle

l = r p

= r m v

NB Only makes sense to talk about the angular momentum wrt or about a certain point

Wed. March 9th 20

Angular Momentum Definition for a Rigid Body About a Fixed Axis

L = li

= I

In analogy to P = M v

Wed. March 9th 21

Newton II for Rotation

• F = dp/dt ( the general form of Newton II)

• = dl/dt

• It is the net torque that causes changes in angular momentum.

Wed. March 9th 22

Angular Momentum Conservation

• Just as for Newton II for linear motion where if the net force was zero, dp/dt =0 => p conserved.

• For rotational motion, if the net torque is zero, dl/dt =0, so l conserved.

Wed. March 9th 23

Angular Momentum Conservation Problems

Wed. March 9th 24

Relating linear and angular variables

• atan = r

• aR = v2/r = 2 r

• v = rRemarks on units and dimensional analysis