Classical Mechanics Lecture 15

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1 Classical Mechanics Lecture 5 Today s Concepts: a) Parallel Axis Theorem b) Torque & Angular Acceleration Mechanics Lecture 5, Slide

2 Unit 4 Main Points Mechanics Lecture 4, Slide

3 Unit 4 Main Points Mechanics Lecture 4, Slide 3

4 Unit 4 Main Points Mechanics Lecture 4, Slide 4

5 Clicker Question A. B. C. D. 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) M dx 0% 0% 0% 0% B) C) D) dx M M L L M dx dx M L x dx Mechanics Lecture 4, Slide 5

6 Clicker Question A. B. C. D. A mass M is uniformly distributed over the length L of a thin rod. The contribution to the rod s moment of inertia provided by element dx is given by: 0% 0% 0% 0% A) B) C) x x M L M L M L dx dx dx M L x dx Mechanics Lecture 4, Slide 6

7 Clicker Question In both cases shown below a hula hoop with mass M and radius is spun with the same angular velocity about a vertical axis through its center. In Case the plane of the hoop is parallel to the floor and in Case it is perpendicular. In which case does the spinning hoop have the most kinetic energy? A) Case B) Case C) Same Case Case ω Everyone got this right too!!! ω Mechanics Lecture 4, Slide 7

8 Clicker Question In which case does the spinning hoop have the most kinetic energy? A) Case B) Case C) Same Case Case ω K Iω ω A) In case one, more mass is located away from its axis, so it has larger moment of inertia. Therefore it has more kinetic energy. Mechanics Lecture 4, Slide 8

9 CheckPoint A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A) 8 B) C) 6 α 30.3% got this right on first attempt Mechanics Lecture 4, Slide 9

10 After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A) 8 B) C) 6 CheckPoint esponse C) The number of revolutions is proportional to time squared. α 0 0 ) ( t t t α ω θ θ + + t ω 0 ω α 0 ; ω θ ) ( ) ( t t t t t t α α θ θ Mechanics Lecture 4, Slide 0

11 CheckPoint A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c axes is I a, I b, and I c respectively. 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 c 59% got this right on first attempt.. Mechanics Lecture 4, Slide

12 CheckPoint esponse Which of the following orderings is correct? A) I a > I b > I c B) I a > I c > I b a b C) I b > I a > I c c ( ) + m( ) + m(0) 8m I a m ( ) + m( ) + m( ) 3m I b m ( ) + m( 0) + m(0) 4m I c m B) I a 8mr^ I b 3mr^ I c 4mr^ Mechanics Lecture 4, Slide

13 Mechanics Lecture 5, Slide 3

14 Clicker Question A. A disk has a radius. The area of a thin ring inside the disk with radius r and thickness dr is: B. C. A) π r dr B) C) π rdr 4π r 3 dr eview geometry. dr r 0% 0% 0% Mechanics Lecture 4, Slide 4

15 A disk spins at revolutions/sec. What is its period? A) T sec Clicker Question A. 0% 0% 0% B. C. B) T π sec C) T ½ sec Mechanics Lecture 4, Slide 5

16 Clicker Question A. B. C. A disk spins at revolutions/sec. 0% 0% 0% What is its angular velocity? A) ω π rad/sec B) ω π rad/sec C) ω 4π rad/sec Mechanics Lecture 4, Slide 6

17 Moment of Inertia of Sphere Must integrate from z-axis not center of! Mechanics Lecture 4, Slide 7

18 Moment of Inertia of Sphere Mechanics Lecture 4, Slide 8

19 Moment of Inertia of Sphere Mechanics Lecture 4, Slide 9

20 Volume of Sphere ) ( cos sin sin sin V r dr r d d dr r d d I d drd r dv dv V r r π π θ θ θ π ϕ θ θ ϕ ϕ θ θ π π π π π Mechanics Lecture 4, Slide 0

21 (i) (ii) (iii) θ ω t + αt θ ω αt ω 0 + ω ω + α θ 0 Using (ii) α ω t ω 0 Using (i) θ αt Mechanics Lecture 4, Slide

22 (iv) I DISK M (v) K Iω Use (iv) Use (v) Mechanics Lecture 4, Slide

23 (vi) (vii) (viii) (ix) d θ v ω a T α v ω a c Use (viii) Use (ix) Mechanics Lecture 4, Slide 3

24 (vi) (vii) (viii) (ix) d θ v ω a T α v ω a c Use (vii) Use (vi) Mechanics Lecture 4, Slide 4

25 I rod I rod ML 3 I disk M I 5 3 fan I disk + 5I rod M + ML Mechanics Lecture 4, Slide 5

26 ( ) t t t t θ θ α α θ θ α ω θ θ + + f αt ω ω + 0 f fan f I K ω Mechanics Lecture 4, Slide 6

27 i f i f i i f f I K I K ω ω ω ω ω ω 4 t t f f 0 0 ω ω α α ω ω + Mechanics Lecture 4, Slide 7

28 Main Points Mechanics Lecture 5, Slide 8

29 Main Points Mechanics Lecture 5, Slide 9

30 Main Points Mechanics Lecture 5, Slide 30

31 Parallel Axis Theorem Mechanics Lecture 5, Slide 3

32 Parallel Axis Theorem Smallest when D 0 Mechanics Lecture 5, Slide 3

33 Clicker Question A. B. C. D. A solid ball of mass M and radius is connected to a rod of mass m and length L as shown. What is the moment of inertia of this system about an axis perpendicular to the other end of the rod? A) M 5 ML I ml B) C) 5 I M + ml + 5 I M + 3 ml 3 ML M L m D) I ML + ml 3 0% 0% 0% 0% axis Mechanics Lecture 5, Slide 33

34 CheckPoint A ball of mass 3M at x 0 is connected to a ball of mass M at x L by a massless rod. Consider the three rotation axes A, B, and C as shown, all parallel to the y axis. For which rotation axis is the moment of inertia of the object smallest? (It may help you to figure out where the center of mass of the object is.) y A B C x 3M M L 00% got this right!!! 0 L/4 L/ Mechanics Lecture 5, Slide 34

35 ight Hand ule for finding Directions Why do the angular velocity and acceleration point perpendicular to the plane of rotation? Mechanics Lecture 5, Slide 36

36 Clicker Question A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular velocity vector point when the ball is rolling up the ramp? A. B. C. D. A) Into the page B) Out of the page C) Up D) Down 0% 0% 0% 0% Mechanics Lecture 5, Slide 37

37 Enter Question Text D. A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling up the ramp? A. B. C. A) Into the page B) Out of the page 0% 0% 0% 0% Mechanics Lecture 5, Slide 38

38 Enter Question Text A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling back down the ramp? A. B. C. D. A) into the page B) out of the page 0% 0% 0% 0% Mechanics Lecture 5, Slide 39

39 Torque τ rf sin(θ ) Mechanics Lecture 5, Slide 40

40 Mechanics Lecture 5, Slide 4

41 CheckPoint In Case, a force F is pushing perpendicular on an object a distance L/ from the rotation axis. In Case the same force is pushing at an angle of 30 degrees a distance L from the axis. In which case is the torque due to the force about the rotation axis biggest? A) Case B) Case C) Same axis axis L/ 90 o F L 30 o F 00% got this right Case Case Mechanics Lecture 5, Slide 4

42 In which case is the torque due to the force about the rotation axis biggest? A) Case B) Case C) Same A) Perpendicular force means more torque. B) F * L torque. L is bigger in Case and the force is the same. CheckPoint axis L/ 90 o Case axis F C) Fsin30 is F/ and its radius is L so it is FL/ which is the same as the other one as it is FL/. L Case 30 o F Mechanics Lecture 5, Slide 43

43 Torque and Acceleration otational nd law Mechanics Lecture 5, Slide 44

44 Similarity to D motion Mechanics Lecture 5, Slide 45

45 Summary : Torque and otational nd law Mechanics Lecture 5, Slide 46

46 Clicker Question Strings are wrapped around the circumference of two solid disks and pulled with identical forces. Disk has a bigger radius, but both have the same moment of inertia. A. B. C. Which disk has the biggest angular acceleration? A) Disk ω ω B) Disk 0% 0% 0% C) same F F Mechanics Lecture 5, Slide 47

47 Clicker/Checkpoint Two hoops can rotate freely about fixed axles through their centers. The hoops have the same mass, but one has twice the radius of the other. Forces F and F are applied as shown. A. B. C. How are the magnitudes of the two forces related if the angular acceleration of the two hoops is the same? A) F F B) F F F F 0% 0% 0% C) F 4F Case Case Mechanics Lecture 5, Slide 48

48 CheckPoint How are the magnitudes of the two forces related if the angular acceleration of the two hoops is the same? A) F F F F B) F F C) F 4F M, Case Case M, B) twice the radius means 4 times the moment of inertia, thus 4 times the torque required. But twice the radiustwice the torque for same force. 4t F x Mechanics Lecture 5, Slide 49

49 τ F sinθ θ 0 θ 90 0 θ 0 θ Mechanics Lecture 5, Slide 50

50 τ F sinθ Direction is perpendicular to both and F, given by the right hand rule τ x τ y 0 0 τ τ + τ + τ z F F F3 Mechanics Lecture 5, Slide 5

51 (i) (ii) I DISK τ Iα M (iii) K Iω Use (i) & (ii) Use (iii) Mechanics Lecture 5, Slide 5

52 Moment of Inertia ) ( ) 5 7 (6 ) ( 5) ( ) (5 5 ) (4 5 ) ( 5 3 total total total rod rod rod total rod total m I m I m m m I L m m L m I I I I Mechanics Lecture 5, Slide 53

53 Moment of Inertia τ rf sinθ τ I α τ α I 7 (6 5 L rod ) m L F sin 90 rod F ( ) Mechanics Lecture 5, Slide 54

54 Moment of Inertia,, ) ( ) ( 5 ) 5 ( 5 ) (4 60 ) ( 5 ) 5 ( 4.5 total total rod rod rod total rod total rod CM rod CM rod CM m I m m m m I m m m L m I I I I m m m m Mechanics Lecture 5, Slide 55

55 Moment of Inertia ) ( sin 0 5 sin m I I F rf τ α α τ θ τ Mechanics Lecture 5, Slide 56

56 Moment of Inertia I I total total I rod + I m rod L rod + m rod (4 ) + 5 m + m ( ) Mechanics Lecture 5, Slide 57

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