Classical Mechanics Lecture 14
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- Basil Strickland
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1 Classical Mechanics Lecture 14 Today s Concepts: a) Rota2onal Mo2on b) Moment of Iner2a Mechanics Lecture 14, Slide 1
2 Will we have to memorize all the different formulas for moment of iner2a? What is a moment of iner2a? Can we go over the equa2ons more thoroughly such as kine2c energy? Will we be required to memorize all the frac2ons of different objects' moment of iner2a, or are we expected to be able to derive the integral? A review of all the concepts would be nice, especially the moment of inter2a. And some homework problems as well please But the bugs on the first ques2on were a bit too realis2c looking. How about a cute, non threatening example next 2me? It was ALL so very confusing!! :'( Your Comments "oooh pick me pick me pick me!"- Donkey from Shrek. Actually, Where did the constants of propor2onality come from when going from a solid cylinder to a cylindrical shell? also, can we do more find the moment of iner2a examples? Mechanics Lecture 14, Slide 2
3 Review What is a Radian? A) A measure of angle B) The distance from the centre of a circle to its circumference C) A brand of space heater D) Something like a jelly fish
4 Review What are the units of a Radian? A) m B) m 1 C) m/degree D) degree/m E) none
5 Review About how large is a radian in degrees? A) 0.6 B) 6.0 C) 60 D) 360 E) 600
6 Summary of Rotations Angular velocity ω is measured in radians/sec Frequency f is measured in revolu2ons/sec 1 revolu2on = 2π radians Period T = 1/f Mechanics Lecture 14, Slide 3
7 Another Summary ω Constant α does not mean constant ω Mechanics Lecture 14, Slide 4
8 Clicker Question A disk spins at 2 revolu2ons/sec. What is its period? A) T = 2 sec B) T = 2π sec C) T = ½ sec Mechanics Lecture 14, Slide 5
9 Clicker Question A disk spins at 2 revolu2ons/sec. What is its angular velocity? A) rad/sec B) rad/sec C) rad/sec Mechanics Lecture 14, Slide 6
10 CheckPoint A wheel which is ini2ally at rest starts to turn with a constant angular accelera2on. Acer 4 seconds it has made 4 complete revolu2ons. How many revolu2ons has it made acer 8 seconds? A) 8 B) 12 C) 16 α Only about half got this right so let s try again Mechanics Lecture 14, Slide 7
11 CheckPoint Response Acer 4 seconds it has made 4 complete revolu2ons. How many revolu2ons has it made acer 8 seconds? A) 8 B) 12 C) 16 α A) Since it made 4 revolu2ons in 4 seconds, its angular velocity is 1 revolu2on per second. Therefore, in 8 seconds, it will have made 8 revolu2ons. B) it makes 4 in the first 4 seconds and then in the second 4 seconds = 12 C) The number of revolu2ons is propor2onal to 2me squared. Mechanics Lecture 14, Slide 8
12 Calculating Moment of Inertia Depends on rota2on axis Mechanics Lecture 14, Slide 9
13 CheckPoint A triangular shape is made from iden2cal balls and iden2cal rigid, massless rods as shown. The moment of iner2a about the a, b, and c axes is I a, I b, and I c respec2vely. Which of the following orderings is correct? A) I a > I b > I c B) I a > I c > I b C) I b > I a > I c a b Only about half got this right so let s try again c Mechanics Lecture 14, Slide 10
14 CheckPoint Response Which of the following orderings is correct? A) I a > I b > I c B) I a > I c > I b C) I b > I a > I c A) I a = 8mr 2 I b = 3mr 2 I c = 2mr 2 B) I a = 8mr 2 I b = 3mr 2 I c = 4mr 2 a r b r c Mechanics Lecture 14, Slide 11
15 Calculation Moment of Inertia Bigger when the mass is further out Mechanics Lecture 14, Slide 12
16 Clicker Question In both cases shown below a hula hoop with mass M and radius R is spun with the same angular velocity about a ver2cal axis through its center. In Case 1 the plane of the hoop is parallel to the floor and in Case 2 it is perpendicular. In which case does the spinning hoop have the most kine2c energy? A) Case 1 B) Case 2 C) Same Case 1 Case 2 R R ω Only about half got this right so let s try again ω Mechanics Lecture 14, Slide 13
17 Clicker Question In which case does the spinning hoop have the most kine2c energy? A) Case 1 B) Case 2 C) Same Case 1 ω R Case 2 A) In case one, more mass is located away from its axis, so it has larger moment of iner2a. Therefore it has more kine2c energy. C) The radius, angular veloci2es and masses are the same so the iner2a is the same which means the kine2c energy is the same for both. ω R Mechanics Lecture 14, Slide 14
18 Clicker Question A mass M is uniformly distributed over the length L of a thin rod. The mass inside a short element dx is given by: A) B) C) D) M analyze the dimensions! L x dx Mechanics Lecture 14, Slide 15
19 Clicker Question A mass M is uniformly distributed over the length L of a thin rod. The contribu2on to the rod s moment of iner2a provided by element dx is given by: A) B) C) M L x dx Mechanics Lecture 14, Slide 16
20 Clicker Question A disk has a radius R. The area of a thin ring inside the disk with radius r and thickness dr is: A) B) C) analyze the dimensions! dr r Mechanics Lecture 14, Slide 17
21 (i) (ii) (iii) Using (ii) Using (i) Mechanics Lecture 14, Slide 18
22 (iv) (v) Use (iv) Use (v) Mechanics Lecture 14, Slide 19
23 (vi) (vii) (viii) (ix) Use (viii) Use (ix) Mechanics Lecture 14, Slide 20
24 (vi) (vii) (viii) (ix) Use (vii) Use (vi) Mechanics Lecture 14, Slide 21
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