Lecture 14 Feb

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1 Lecture 14 Feb Moving in circles Angular momentum Spinning at the Olympics Torques Why is a wrench useful? Center of Gravity Useful info for crossing Niagara falls on a wire The New Hubble Telescope Views of Saturn 2/12/2016 Physics 214 Spring

2 Balancing a Balance Construct a balance of a thin but rigid and uniform beam, supported by a wedge (fulcrum) m 1 m2 Table We test the reliability of the instrument by balancing it without any weight. We find that equal weights located at equal distances will balance. By balancing we mean that it will not tend to rotate about the fulcrum. 2/12/2016 Physics 214 Spring

3 Suppose we try balancing unequal weights. Clearly at equal distances from the fulcrum there is no balance. If however the smaller weight is further from the fulcrum such that m l 1 1 m2l2 Eq.1 they will balance. This will be true for many combination of m1 l1 m2. l2. To understand the physical meaning of Eq.1 clearer, multiply both side by g and denote the distances by r Thus we obtain: Note that m g and of the two masses. m g 1 2 m gr 1 1 m2gr2 Eq.2 are both force like quantities, the weights 2/12/2016 Physics 214 Spring

4 Thus what we are seeking is equality of: Force distance = torque F r Force vector arm vector torque vector Note that here the arm vectors and force vectors are perpendicular to each other. 2/12/2016 Physics 214 Spring

5 In studying work as F r F r cos that is the scalar product of force distance cos where is the angle between them. To describe the effect of torque we have to study the way torque acts on a wheel. r F F F If the force acts along or opposite to r nothing move. But if the force normal to r, starts the wheel rotating. Thus a new kind of vector product is introduced. r F r F sin When = 0 or 180, sin = 0 thus = 0. At = 90 sin = 1 and acts most effectively. 2/12/2016 Physics 214 Spring

6 Using an adjustable Wrench Adjust the wrench to hold the nut. The arm of the handle of the wrench is the lever arm of the force l and hand provide the force. If the nut does not move, increase the size of the wrench handle thus the size of l. 2/12/2016 Physics 214 Spring

Forces and torques If we apply a force to a bicycle wheel that is free to rotate for a given force it is easier to rotate the wheel the further you are from the axle.

7 Forces and torques If we apply a force to a bicycle wheel that is free to rotate for a given force it is easier to rotate the wheel the further you are from the axle. In the picture shown below each of the single weights on it s own will cause the rule to rotate but the two together can be balanced. A force applied to an object, in general makes that object rotate and the action of the force we call a TORQUE and Τ = FL where L is the perpendicular distance to the line of action of the force. Once again + is counterclockwise and is clockwise and the net torque is sum of all torques. 2/12/2016 Physics 214 Spring

8 Using a wrench In our everyday life we are limited in the force we can apply but if we increase the lever arm we can increase the torque. We can turn a very tight nut by applying a force F at a large radius R If the radius of the nut is r then to just move the nut FR = F f r so if R/r = 30 then F s /F = 30 The work done is the same because in one turn 2πRF = 2πrF s R F f F 2/12/2016 Physics 214 Spring

9 1J-20 Torque Wrench Torque wrench This tool is used to tighten a nut to an exact value. Usually used in industry like car manufacture or repair 2/12/2016 Physics 214 Spring

10 Rotation Dynamics Newton s Second Law for Rotational Motion About a Fixed Axis Consider a particle moving on a circular path like a model plane on a guideline. The plane s engine produces a net external tangential force F that gives a plane a tangential acceleration a t. In accord with Newton s second law it follows that The torque produced by this force is r F t ma r F t Where radius of the circular path is also the lever arm. Thus the torque in terms of m, a and r is mr a t t t 2/12/2016 Physics 214 Spring

11 a t While is different at every point of the solid body the angular acceleration is the same everywhere in a solid body and we can express anywhere as at r Thus we obtain for the magnitude of mr 2 The proportionality constant between the torque and the 2 angular acceleration is " mr " which is called the moment of inertia of a mass m which is located at perpendicular distance r from the rotation axis. Summing all mass points of the solid body, finally we have a t 2 i net ( mr i i ) Net external torque Moment of inertia of solid body 2/12/2016 Physics 214 Spring

12 I denotes the inertia of the solid body. The kinetic energy of the solid body is : EK 1 2 I 2 Where is the angular velocity of the solid body. The angular momentum L of the solid body is L I 2/12/2016 Physics 214 Spring

13 Moment of Inertia The Moment of Inertia is always of the form I = mass times a length squared and it depends on the distribution of mass about the axis of rotation 2/12/2016 Physics 214 Spring

14 Conservation of angular momentum Linear momentum for a particle P = mv For a rotating object L = Iω is the angular momentum Angular momentum is conserved in a closed system. That is one in which no external torques are acting. A simple closed system is a skater and if I is changed ω will to keep L at the same value. We can invert the bicycle wheel and ω will change to keep L constant L = I ω 2/12/2016 Physics 214 Spring

1Q-20 Conservation of angular momentum Changing the moment of inertia for a closed system What happens when we pull the cord so that the two spheres come closer together?

There is no net torque (the forces are internal)so L is a constant but the moment of inertia changes and since L = Iω as I decreases ω increases.

15 1Q-20 Conservation of angular momentum Changing the moment of inertia for a closed system What happens when we pull the cord so that the two spheres come closer together? The dominant physical law is conservation of angular momentum. There is no net torque (the forces are internal)so L is a constant but the moment of inertia changes and since L = Iω as I decreases ω increases. So if I decreases by a factor of 2 then ω increases by a factor of 2. The kinetic energy is = Iω 2 so if I decreases the kinetic energy increases by the amount of work Fd where F is the force applied to the cord and d is the distance moved. So in the above example the kinetic energy increases by a factor of 2. It requires significant force for a skater to pull in his arms. 2/12/2016 Physics 214 Spring

torque is small The torque causes the vector L to precess and changes the direction of the angular momentum

16 1Q-30 Bicycle Wheel Gyroscope Gyroscopic action and precession L What happens to the wheel, does it fall down? F = mg mg F The counterclockwise torque adds to L and produces a precession, providing L is large and the torque is small The torque causes the vector L to precess and changes the direction of the angular momentum vector which is perpendicular to the plane of rotation. This is a very large top. 2/12/2016 Physics 214 Spring

17 1Q-21 Conservation of angular momentum Conservation of angular momentum using a spinning wheel What happens when the wheel is inverted? The dominant physical law is conservation of angular momentum. To change the angular momentum of the wheel requires an external torque. So although we can change the direction of the angular momentum of the wheel the force we use is internal to the wheel/stool system so the the stool rotates to keep the net angular momentum the same To turn the wheel requires significant force and work is needed. The energy of the final system is greater than the initial energy by the amount of work that is done. 2/12/2016 Physics 214 Spring

1Q-32 Stability Under Rotation Example of Gyroscopic Stability: Swinging a spinning Record Why does the Record not flop around once it is set spinning?

18 1Q-32 Stability Under Rotation Example of Gyroscopic Stability: Swinging a spinning Record Why does the Record not flop around once it is set spinning? L L The dominant physical law is conservation of angular momentum. With no torque the vector L, perpendicular to the plane of rotation always points in the same direction. SINCE THERE IS NO TORQUE ABOUT THE CENTER OF ROTATION OF THE RECORD, THE ANGULAR MOMENTUM VECTOR CANNOT CHANGE. THIS IS GYROSCOPIC STABILITY. THIS IS A VERY SIMPLE GYROSCOPE AND SOPHISTICATED GYROSCOPES ARE USED TO STEER AIRCRAFT AND ORIENT THE HUBBLE TELESCOPE 2/12/2016 Physics 214 Spring

19 1Q- 23 Conservation of angular momentum Changing the moment of inertia of a skater How does conservation of angular momentum manifest itself? This is two examples of Conservation of angular momentum The first changes the Moment of Inertia (like a skater) The second shows what happens when you swing a bat. All forces are internal to the system so L is conserved. Case 1 the moment of inertia changes and since l = Iω the speed of rotation changes to keep L constant. Case 2 since L = 0 swinging the bat causes the person to rotate in the opposite direction. 2/12/2016 Physics 214 Spring

20 When an object rolls and slides down a slope the acceleration and final speed depend on the Moment of Inertia.This is because the energy is divided between translation and rotation Mgh = 1/2Mv 2 + 1/2Iω 2 If there is no sliding the motion is pure rotation and Mgh = 1/2Iω 2 With ω = v/r For a disk (I = 1/2 MR 2 ) and v = sqrt(4/3gh) but for a hoop (I = MR 2 ) v = sqrt(gh) So the disk is faster at the bottom 1Q-04 Translation with Rolling 2/12/2016 Physics 214 Spring

Rotating object To describe the properties of a rotating object I is the moment of inertia which depends on the distribution of mass about the axis of rotation.

21 Rotating object To describe the properties of a rotating object I is the moment of inertia which depends on the distribution of mass about the axis of rotation. The kinetic energy = ½ Iω 2 Angular momentum L = Iω is a vector perpendicular to the rotation plane Rotation is changed by torque T = Iα (+ counter clockwise) Closed system angular momentum is conserved L = I ω. What about energy and work Kinetic energy = ½ Iω 2 Suppose I changes by a factor of 3 smaller Then since L is conserved the ω new = 3 ω old KE new = 3 KE old this energy comes from the work done internally to change I. The skater needs to exert a force to pull her arms in to her body. 2/12/2016 Physics 214 Spring

22 Summary of Chapter 8 + Rotational motion Angular velocity ω = Δθ/Δt Angular acceleration α = Δω/Δt One full circle = = 2π radians Circumference = 2πR Time for one revolution = 2πR/v 2πr/v = 2π/ω and v = rω v R Displacement d θ Velocity v = Δd/Δt ω = Δθ/Δt Acceleration a = Δv/Δt α = Δω/Δt Constant v = v 0 + at ω = ω 0 + αt d = v 0 t + 1/2at 2 θ = ω 0 t + 1/2αt 2 v 2 = v ad ω 2 = ω αθ d = ½(v + v 0 )t ω = ½(ω + ω 0 2/12/2016 Physics 214 Spring

Torques Torque = FL where where L is the perpendicular distance to the line of action of the force + is counterclockwise and is clockwise net torque is sum of all torques For the boy on the plank he

23 Torques Torque = FL where where L is the perpendicular distance to the line of action of the force + is counterclockwise and is clockwise net torque is sum of all torques For the boy on the plank he will fall when W p d p > W c d c Torque = FL = Iα I plays the role of mass for rotation Kinetic energy = 1/2Iω 2 Work = Tθ ( full circle T2π = Fr2π Angular momentum L = Iω There is a point in the geometry of a body at which all the mass appears to act and one can balance the body with a single force (center of mass/gravity) g 2/12/2016 Physics 214 Spring

24 Conservation of angular momentum In a closed system L = Iω we can change I and ω will change like a spinning skater. We can invert the bicycle wheel and ω will change to keep L constant The kinetic energy changes because the person does work and has to exert force in order to change I 2/12/2016 Physics 214 Spring

The New Hubble http://hubblesite.org/ These four images are among the first observations made by the new Wide Field Camera 3 aboard the upgraded NASA Hubble Space Telescope.

25 The New Hubble These four images are among the first observations made by the new Wide Field Camera 3 aboard the upgraded NASA Hubble Space Telescope. The image at top left shows NGC 6302, a butterfly-shaped nebula surrounding a dying star. At top right is a picture of a clash among members of a galactic grouping called Stephan's Quintet. The image at bottom left gives viewers a panoramic portrait of a colorful assortment of 100,000 stars residing in the crowded core of Omega Centauri, a giant globular cluster. At bottom right, an eerie pillar of star birth in the Carina Nebula rises from a sea of greenish-colored 2/12/2016 Physics 214 Spring

Saturn 1610 - Galileo Galilei becomes the first to observe Saturn's rings with his 20-power telescope.

In this view, the giant orange moon Titan casts a large shadow onto Saturn's north polar hood.

Farther to the left, and off Saturn's disk, are the bright moon Dione and the fainter moon Enceladus.

26 Saturn Galileo Galilei becomes the first to observe Saturn's rings with his 20-power telescope. On February 24, 2009, the Hubble Space Telescope took a photo of four moons of Saturn passing in front of their parent planet. In this view, the giant orange moon Titan casts a large shadow onto Saturn's north polar hood. Below Titan, near the ring plane and to the left is the moon Mimas, casting a much smaller shadow onto Saturn's equatorial cloud tops. Farther to the left, and off Saturn's disk, are the bright moon Dione and the fainter moon Enceladus. These rare moon transits only happen when the tilt of Saturn's ring plane is nearly "edge on" as seen from the Earth. Saturn's rings were perfectly on edge to our line of sight on August 10, 2009, and September 4, Unfortunately, Saturn was too close to the sun to be seen by viewers on Earth at that time. This "ring plane crossing" occurs every years. In Hubble witnessed the ring plane crossing event, as well as many moon transits, and even helped discover several new moons of Saturn. Scientists at NASA have discovered a nearly invisible ring around Saturn -- one so large that it would take 1 billion Earths to fill it. The ring's orbit is tilted 27 degrees from the planet's main ring plane. The bulk of it starts about 3.7 million miles (6 million km) away from the planet and extends outward another 7.4 million miles (12 million km). 2/12/2016 Physics 214 Spring

27 Questions Chapter 8 Q6 Is the linear speed of a child sitting near the center of a rotating merry-go-round the same as that of another child sitting near the edge of the same merry-go-round? Explain. The angular velocity is the same but v = ωr, so speed is greatest at the edge Q11 The two forces in the diagram have the same magnitude. Which orientation will produce the greater torque on the wheel? Explain. F 1 F 2 F 1 because it is the tangential component that produces the torque 2/12/2016 Physics 214 Spring

28 Q13 Is it possible for the net force acting on an object to be zero, but the net torque to be greater than zero? Explain. (Hint: The forces contributing to the net force may not lie along the same line.) F F Q20 Two objects have the same total mass, but object A has its mass concentrated closer to the axis of rotation than object B. Which object will be easier to set into rotational motion? Explain. A has a smaller moment of inertia and torque = Iα so will accelerate faster 2/12/2016 Physics 214 Spring

29 Q26 A child on a freely rotating merry-go-round moves from near the center to the edge. Will the rotational velocity of the merrygo-round increase, decrease, or not change at all? Explain. L =Iω so I increases and ω will decrease. This requires work Q29 Suppose you are rotating a ball attached to a string in a circle. If you allow the string to wrap around your finger, does the rotational velocity of the ball change as the string shortens? Explain. L =Iω so I decreases and ω will increase. This requires work 2/12/2016 Physics 214 Spring

30 Ch 8 E 2 Rate of rotation = 45 RPM (rev/min) a) What is this in rev/s? b) How many revolutions in 5 seconds? a) 45 RPM = 45 rev/min 1 min/60sec = ¾ rev/sec b) ¾ rev/s (5s) = 15/4 rev 2/12/2016 Physics 214 Spring

31 Ch 8 E 6 Rotational velocity decreases from 6 rev/s to 3 rev/s in 12 s. What is rotational acceleration? w = w f w i = 3rev/s 6rev/s = - 3rev/s = w/t = (-3rev/s)/12s = -1/4 rev/s 2 = -1/4 rev/s 2 rad/rev = - /2 rad/s 2 2/12/2016 Physics 214 Spring

32 Ch 8 CP 10 5N placed 10 cm from fulcrum of balance beam, what weight should be put 4 cm from fulcrum on other side to balance = Fl = 0 = -(5N)(10cm) + x(4cm) x = 50Ncm/4cm = 12.5 N? 5N 4cm 10cm 2/12/2016 Physics 214 Spring

33 Ch 8 E kg mass rotating with light, rigid 50 cm rod with w = 3 rad/s. a) What is rotational inertia? b) What is angular momentum? a) I = mr 2 = (0.8 kg)(0.50m) 2 = 0.2kg m 2 W = 3 rad/s b) L = Iw = (0.2 kg m 2 )(3 rad/s) = 0.6kg m 2 /s M 50cm 2/12/2016 Physics 214 Spring

34 Ch 8 CP 2 4m plank, weight = 80N. Pivot = 1m from far end. 150N weight on pivot and moving slowly outward (toward far end). a) What torque is exerted by plank s weight about pivot? b) How far can 150N weight move? c) How can you test (b) without flipping the plank? 4m 2m 1m 1m a) 1 = Fl = (80N)(1m) = +80Nm b) = 0 = 1 (150N)x, x = 80Nm/150N = 0.53m c) Think about it! W = 80N W = 150N Far End 2/12/2016 Physics 214 Spring

35 Ch 8 CP 4 See Fig A student sits on a stool with wheels with a bike wheel with I b = 2kgm 2 and w b = 5 rev/s. Bike wheel is spinning, student is not. The student on stool with bike wheel: I s = 6 kgm 2. a) Initially, what is L? b) Student flips bike wheel. What is student s L? c) Where does the torque come from that accelerates student? a) L b = I b w b (just the bike wheel spinning). w b = 5 rev/s 2 /rev = 10 rad/s L b = 20 kg m 2 /s, upwards b) L = L b + L s (flipped bike wheel plus student). -20 kg m 2 /s + I s w s = +20 kg m 2 /s L s = +40 kg m 2 /s c) Student supplies torque when he flips bike wheel. 2/12/2016 Physics 214 Spring

7 th week Lectures Feb

7 th week Lectures Feb.19. 23. 2018. Moving in circles Torques Why is a wrench useful? Angular momentum Spinning at the Olympics Center of Gravity Useful info for crossing Niagara falls on a wire The New