Wed. March 9th1 PHSX213 class Class stuff –HW6W returned. –HW7. Worth 1.5 times normal...
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Transcript of Wed. March 9th1 PHSX213 class Class stuff –HW6W returned. –HW7. Worth 1.5 times normal...
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 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 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 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.