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Physics 7B - AB Lecture 7 May 15 Recap Angular Momentum Model (Second half of Chap 7) Recap Torque, Angular Momentum Rotational Inertia (new concept!) Intro to Newtonian Model (start Chapter 8) 1 Re-evaluation Request Due Quiz 2 May 22 (next Thursday) Quiz 3 May 29 Quiz 3 average 7.75 (C+) Quiz 4 graded, solution up on the web, rubrics will follow 2 Recap Rotational (Angular) Motion 1. A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular speed, , is is gentleman bug z A) half the ladybug’s. B) the same as the ladybug’s. C) twice the ladybug’s. D) impossible to determine. y x Gentleman bug Ladybug 3 Recap Rotational (Angular) Motion 1. A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular speed, , is is gentleman bug z A) half the ladybug’s. B) the same as the ladybug’s. C) twice the ladybug’s. D) impossible to determine. y x Gentleman bug Ladybug B) The two bugs each travel the same angle (ie. one revolution or 2p radians or any other angle) in the same amount of time so they have the same angular velocity 4 Recap Rotational (Angular) Motion 2. The ladybug’s angular velocity vector, , points along ladybug points along the the A) B) C) D) E) + + + x-axis x-axis y-axis y-axis z-axis z y x Ladybug 5 Recap Rotational (Angular) Motion 2. The ladybug’s angular velocity vector, , points along ladybug points along the the A) B) C) D) E) + + + x-axis x-axis y-axis y-axis z-axis z y x Ladybug E) Use your right hand to show that the angular velocity points along the +z axis. 6 Recap Rotational (Angular) Motion 3. The wheel shown to the right can rotate freely about its axle (the dot in the center) which is fixed in space like the wheel in lecture. A force, F, is applied as shown F (along with whatever other forces are acting on it). F produces a torque aboutthethe of mass. pivotcenter point. is the direction of torque this torque? What is the direction of this ? What F A) Clockwise away rotation) from you). A) Torque vector(torque pointingvector into thepointing slide (clockwise B) Counterclockwise (torque vector pointing toward you). B) Torque vector pointing out of the slide (counterclockwise rotation) C) Torque is zero. C) Torque is zero. 7 Recap Rotational (Angular) Motion 3. The wheel shown to the right can rotate freely about its axle (the dot in the center) which is fixed in space like the wheel in lecture. A force, F, is applied as shown F (along with whatever other forces are acting on it). F produces a torque aboutthethe of mass. pivotcenter point. is the direction of torque this torque? What is the direction of this ? What F A) Clockwise away rotation) from you). A) Torque vector(torque pointingvector into thepointing slide (clockwise B) Counterclockwise (torque vector pointing toward you). B) Torque vector pointing out of the slide (counterclockwise rotation) C) Torque is zero. C) Torque is zero. B) You should see that pushing this way will tend to rotate the wheel counterclockwise about its axle and that the right hand rule give you a torque vector pointing toward you. 8 Recap Rotational (Angular) Motion 4. The picture to the right shows the vtablecloth famous “yank the tablecloth out from vcloth under the wine goblet” trick. The tablecloth is pulled to the right as shown. The forces on the goblet are shown below the picture. The friction force by the tablecloth produces a torque about the center of mass. What direction is this torque? A) Torque vector due to friction force points Fgravity by earth away from you. FEarth on the goblet on goblet B) Torque vector due to Ffriction F / /tablebycloth on the goblet tablecloth friction force points on goblet Fnormal by tablecloth toward you. (friction) F table cloth on the goblet on goblet C) Torque is zero. 9 Recap Rotational (Angular) Motion 4. The picture to the right shows the vtablecloth famous “yank the tablecloth out from vcloth under the wine goblet” trick. The tablecloth is pulled to the right as shown. The forces on the goblet are shown below the picture. The friction force by the tablecloth produces a torque about the center of mass. What direction is this torque? A) Torque vector due to friction force points Fgravity by earth away from you. FEarth on the goblet on goblet B) Torque vector due to Ffriction F / /tablebycloth on the goblet tablecloth friction force points on goblet Fnormal by tablecloth toward you. (friction) F table cloth on the goblet on goblet C) Torque is zero. B) Very similar to the previous question. If the goblet tips over, it will tip over counterclockwise. 10 Torque - rotational force that can change the rotational motion Force is exerted tangentially on the rim, the rim is at a distance r (moment arm) from the pivot point. Ftangential r Direction of Torque Force and Torque are two different physical quantities! Direction of is given by the RHR Torque this way Rotation this way 11 Rotational motion is changed by applying forces, But where the force is applied is Just as important as the size of the force Ftangential r Magnitude of Torque = r Ftangential = r Ftangential 12 Force and Torque are two different physical quantities! Recap Extended Force Diagram L/4 5. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. downward force, F Fby string on plank, be placed? Where should the string on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 13 Recap Extended Force Diagram L/4 5. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. downward force, F Fby string on plank, be placed? Where should the string on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 14 Recap Extended Force Diagram L/4 6. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. Where should the upward force,F Fby fulcrum on plank, be placed? fulcrum on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 15 Recap Extended Force Diagram L/4 6. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. Where should the upward force, FFby fulcrum on plank, be placed? fulcrum on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 16 Recap Extended Force Diagram L/4 7. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. downward force, FFby earth on plank, be placed? Where should the Earth on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 17 Recap Extended Force Diagram L/4 7. The uniform plank shown to the right has a length, L, and a mass M. The plank rests on a fulcrum and a chunk of metal with mass M/2 hangs by a string from the left end. downward force, FFby earth on plank, be placed? Where should the Earth on Plank A) B) C) D) E) At the left end where L/4 from the left end L/2 from the left end At the right end. The force is equal to the string is tied. where the fulcrum is. where the center of mass is. zero so it doesn’t matter. 18 Recap Rotational (Angular) Motion L/4 8. The uniform plank shown to the right has a length, L, and a mass The weight hanging from the M. left end of the plank has a mass In order to equal to M/2. balance the plank as shown, a force has been exerted on the right end of the plank. How large is this force and what direction is it in? A) The force is upwards. B) The force is upwards. C) The force is downwards. D) The force is downwards. E) The force is bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted equal to zero. 19 Ffulcrum on Plank F ?? L/4 8. The uniform plank shown to the right has a length, L, and a mass The weight hanging from the M. left end of the plank has a mass In order to equal to M/2. the has been exerted force balance the plank as shown, Fstring onaPlank = – (M/2)g FEarth ononPlank = –right Mg end of the plank. How large is this force and what direction is it in? A) The force is upwards. B) The force is upwards. C) The force is downwards. D) The force is downwards. E) The force is bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted equal to zero. 20 Angular analogue to Impulse is Angular Impulse • Angular Impulse Is related to the net external torque in the following way: Net Angular Impulseext = ∆ L = ∫ ext(t)dt If the torque is constant during a time interval ∆t Net Angular Impulseext = ∆ L = ave.ext x ∆ t If the net torque is zero, the plank will stay stationary… 21 Ffulcrum on Plank F L/4 8. The uniform plank shown to the right has a length, L, and a mass The weight hanging from the M. left end of the plank has a mass In order to equal to M/2. the has been exerted force balance the plank as shown, Fstring onaPlank = – (M/2)g FEarth ononPlank = –right Mg end of the plank. How large is this force and what direction is it in? A) The force is upwards. B) The force is upwards. C) The force is downwards. D) The force is downwards. E) The force is bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted equal to zero. 22 Ffulcrum on Plank F = (M/6)g L/4 8. The uniform plank shown to the right has a length, L, and a mass The weight hanging from the M. left end of the plank has a mass In order to equal to M/2. the has been exerted force balance the plank as shown, Fstring onaPlank = – (M/2)g FEarth ononPlank = –right Mg end of the plank. How large is this force and what direction is it in? A) The force is upwards. B) The force is upwards. C) The force is downwards. D) The force is downwards. E) The force is bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted bigger than the weight on the left and it is exerted smaller than the weight on the left and it is exerted equal to zero. ave.ext = (L/4)(M/2)g + (0)(M/2)g – (L/4)Mg + (3L/4) F = 0 (L/4)(M/2)g – (L/4)Mg + (3L/4) F = 0 (3L/4) F = (L/4)(M/2)g 3 F = (M/2)g F = (M/6)g 23 Define Angular Momentum L Think of rotational inertia I kind of like mass for now. rotating like this with Magnitude of Angular Momentum L=I 24 Define Angular Momentum L Think of rotational inertia I kind of like mass for now. Direction of L is given by the RHR rotating like this with Magnitude of Angular Momentum L=I Rotation This way 25 What do you mean by rotational analog to mass? Rotational Inertia Depends not only on the amount of mass in the object but also on how the mass is distributed about the axis of rotation : I can change! Formula not really important, but the idea is that the further mass is from the axis of rotation, the greater I Example: Same mass, same volume but arranged differently r2 r1 rotates this way I1 rotates this way > I2 26 What do you mean by rotational analog to mass? Rotational Inertia the idea is that the further mass is from the axis of rotation, the greater I Point mass m r Thin ring of mass m r Disk of mass m r 27 What do you mean by rotational analog to mass? Rotational Inertia the idea is that the further mass is from the axis of rotation, the greater I Point mass m r Thin ring of mass m r Disk of mass m r I = mr2 28 What do you mean by rotational analog to mass? Rotational Inertia the idea is that the further mass is from the axis of rotation, the greater I Point mass m r I = mr2 Thin ring of mass m r Disk of mass m r I = mr2 29 What do you mean by rotational analog to mass? Rotational Inertia the idea is that the further mass is from the axis of rotation, the greater I Point mass m r I = mr2 Thin ring of mass m r I = mr2 Disk of mass m r I = (1/2)mr2 30 Question 31 Question 32 Why does a figure skater start spinning faster when she pulls her arms in? Initial initial Final < final 33 Assume ice surface is frictionless, the net torque on the skater is zero… Net Angular Impulseext = ∆ L = ave.ext x ∆ t = 0 Angular Momentum Lskater is conserved! Initial L initial, skater = I initial, skater initial Final L final, skater = I final, skater final I initial, skater initial = I final, skater final Remember I initial, skater > I final, skater So when the rotational inertia decreases (which it does when she pulls her arms in), angular velocity must increase in order to conserve the angular momentum Spin control is nothing but invoking Conservation of Angular momentum! 34 Newtonian Model Ouch… Umm why does an apple fall ?? I tried to understand the force on an apple and its relation to apple’s motion. It is all summarized in Newton’s Laws of Motion. Sir. Isaac Newton 35 Newtonian Model Newton’s Laws of Motion Newton’s first law: If the momentum changes, there is a net force on the system. If the momentum is not changing, there is no net force on the system. Net Impulseext = ∆ p = ∑ Fave.ext x ∆ t Newton’s second law: Quantitatively relates instantaneous change in momentum (or velocity) to net force ( ∑ Fave.ext = ∆ p /∆ t ) in terms of instantaneous time rate change of momentum… ∑ Fext = d p /dt = m dv/dt = ma An unbalanced force (∑ Fext 0) causes a change in motion of an 36 object,i.e.time rate change of velocity (acceleration) Newtonian Model Newton’s Laws of Motion Newton’s first law: If the momentum changes, there is a net force on the system. If the momentum is not changing, there is no net force on the system. Net Impulseext = ∆ p = ∑ Fave.ext x ∆ t Newton’s second law: Quantitatively relates instantaneous change in momentum (or velocity) to net force ∑ Fext = d p /dt = m dv/dt = ma An unbalanced force (∑ Fext 0) causes a change in motion of an object,i.e.time rate change of velocity (acceleration) Newton’s third law: F A on B = – F B on A You cannot push without being pushed yourself! 37 When is Newtonian Model valid ? When things are not too small and its motion not too fast Protons + Neutrons Quic kTime™ and a TIFF (Unc ompres sed) decompress or are needed to see this picture. Electrons Air France Concorde Mach 2.23 = 7.58 x 102 m/s << Speed of light (3 x 108 m/s) 38 How is Newtonian Model useful ? We know the net force on the object and want to know what its subsequent motion is OR We know the motion of an object and want to know details of the forces acting on the object We know aconcorde, What is F ejected gas on the concorde? We know the gravitational forces between all the planets39and the sun. When/Where is next total solar eclipse? Practice Newton’s Laws of Motion 1. A) for True or B) for False. A box rests on the floor. The upward force exerted by the floor surface on the box forms a Newton’s 3rd Law pair with the force exerted downward on the box by the Earth’s gravity. B) False! According to Newton’s 3rd Law: Fgravity by earth on box = -Fgravity by box on earth And Fnormal by floor on box = -Fnormal by box on floor 40 Practice Newton’s Laws of Motion 1. A) for True or B) for False. A box rests on the floor. The upward force exerted by the floor surface on the box forms a Newton’s 3rd Law pair with the force exerted downward on the box by the Earth’s gravity. B) False! According to Newton’s 3rd Law: Fgravity by earth on box = -Fgravity by box on earth And Fnormal by floor on box = -Fnormal by box on floor 41 Practice Newton’s Laws of Motion 2. A child sits on a stationary swing held up by chains. The child feels the normal force up from the seat of the swing. Which force is the equal-and-opposite pair to this one, according to Newton's 3rd Law? A) B) C) D) E) Force of child's weight down on chain. Tension force up on seat by chain. Gravity force down on child. Normal force down on seat by child. none of the above. D) According to Newton’s 3rd Law: Fnormal by seat on child = -Fnormal by child on seat 42 Practice Newton’s Laws of Motion 2. A child sits on a stationary swing held up by chains. The child feels the normal force up from the seat of the swing. Which force is the equal-and-opposite pair to this one, according to Newton's 3rd Law? A) B) C) D) E) Force of child's weight down on chain. Tension force up on seat by chain. Gravity force down on child. Normal force down on seat by child. none of the above. D) According to Newton’s 3rd Law: Fnormal by seat on child = -Fnormal by child on seat 43 Position vs Velocity vs Acceleration 3. The graph to the right shows the position as a function of time for two trains running on parallel tracks. Which is true? B A) At time, tB, both trains have the same velocity. B) Both trains speed up all the time. C) Both trains have the same velocity at some time before tB. D) None of the above are true. C) velocity is slope of position graph so both trains have the same velocity at time t1 A position tB t1 time 44 Position vs Velocity vs Acceleration 3. The graph to the right shows the position as a function of time for two trains running on parallel tracks. Which is true? B A) At time, tB, both trains have the same velocity. B) Both trains speed up all the time. C) Both trains have the same velocity at some time before tB. D) None of the above are true. C) velocity is slope of position graph so both trains have the same velocity at time t1 A position tB t1 time 45 Be sure to write your name, ID number & DL section!!!!! 1 MR 10:30-12:50 Dan Phillips 2 TR 2:10-4:30 Abby Shockley 3 TR 4:40-7:00 John Mahoney 4 TR 7:10-9:30 Ryan James 5 TF 8:00-10:20 Ryan James 6 TF 10:30-12:50 John Mahoney 7 W 10:30-12:50 Brandon Bozek 7 F 2:10-4:30 Brandon Bozek 8 MW 8:00-10:20 Brandon Bozek 9 MW 2:10-4:30 Chris Miller 10 MW 4:40-7:00 Marshall Van Zijll 11 MW 7:10-9:30 Marshall Van Zijll 46