1、Brazilian Team,7. Coin,“Stand a coin on its edge upon a horizontal surface. Gently spin the coin and investigate the resulting motion as it settles.”,7.1.1. Rotational Inertia,7.1. Basic Concepts,7.1.2. Angular momentum,7.2. Methodology,Part A:General view of the movementPart B:Theoretical analyses
2、of the movementPart C:Experimental verification of the parameters,7.3. General view,7.3. General view,Motion observed,P,P,Rotation,Precession of the axis,Motion of the center of mass,7.4. Theoretical analyses,7.4.1. Strategy,7.4.2. Coin rotation,y,z,Decreasing angular momentum,7.4.2. Coin rotation,x
3、,x,y,z,F,7.4.3. Precession of the rotating axis part I,axis,7.4.3. Precession of the rotating axis part II,L,As the axis inclinates the torque increases.,precession,7.4.4. Motion of the center of mass,The coin tends to slide. So, a friction force appears as indicated.The contact point will describe
4、a circle.,trajectory,7.4.5. Special coin confirming the accuracy of our analysis,In a normal spining coin the axis is changing all the time,By making this distribution of mass we created a fixed rotating axis.,Black tape,White tape,The axis will inclinate precession of the axis Centripetal force in
5、the direction of the axis,7.4.5. Special coin confirming the accuracy of our analyses,7.4.6. Precession predominates,The angular momentum will decrease The torque due to the binary and will increase Thus the precession velocity (P) will increase Comprovation by the measurement of the frequency of th
6、e sound,7.4.6. Precession predominates,7.5.1. Materials usedOld R$0,05 coin new R$0,50 coin 30cm-ruler chronometer sandpaper wooden floor video camera,7.5 Experimental analyses,7.5. Experimental analyses,7.5.2. ProcedureIn order to analyse the influence of the mass and the friction force, it was use
7、d:Two different coins Three different surfaces,7.5. Experimental analyses,7.5.3. Results: Surface A,7.5. Experimental analyses,7.5.4. Results: Surface B,7.5. Experimental analyses,7.5.5. Results: Surface C,7.5. Experimental analyses,7.5.6. Comparison,7.6. Error analysis,7.6.1. Main source of errorDi
8、fferent forces Coin width Surface inclination Chronometer precision,7.7. Conclusion,Therefore we can conclude that there are three important forces in the movement:Weight Normal Force Friction,7.7. Conclusion,Because of this forces:Curve ray gets smaller Tangent velocity gets smaller Angular velocit
9、y gets smaller Precession velocity gets bigger,7.4. Results,7.4.1. Analysis of the motion in three stages,1st Stage: the translation motion predominates, which means that the tangential speed v is the main component of the motion.,V,7.4. Results,7.4.1. Analysis of the motion in three stages2nd stage
10、: the rotational motion predominates. This means that the centripetal acceleration a, caused by the attrition force, is the main responsible for the curvilinear motion of the coin.,a,7.4. Results,7.4.1. Analysis of the motion in three stages3rd stage: when the coin is stabilized on a certain positio
11、n, it is clearly verified the “spinning top” effect, in which the angular speed causes the coin to spin around its axle, but also has its axle altered by the weight-force, until the coin falls down.,Final Movement,7.4. Results,7.4.2. Analysis of the influence of the mass,It was verified that the hea
12、vier coin (R$0,50) performs its motion in a shorter interval of time. Therefore, since the attrition force is higher, the speed is reduced faster and, because of its weight, the coin falls down faster.,7.4. Results,7.4.3. Analysis of the influence of the surfaceIt was noticed that the higher the att
13、rition coefficient, the higher the centripetal acceleration on the second stage. Consequently, on the third stage, as the coin will be slower, it will fall down more rapidly.,7.6. Parameters,Coin mass Coin shape Coin contact surface Coin ray Local inclination Friction coefficient between materials,7
14、.7. Conclusions,The linear speed v decreases through the trajectory; the centripetal acceleration increases angular acceleration is distinguished on the last stage of the motion,7.7. Conclusions,Thats why the coin moves approximately in a spiral form. This awkward phenomenon happens because “the rol
15、ling coin squeezes and swirls the air beneath. The flowing air takes up energy, tipping the coin even closer to the surface. At some point, the coins edge finally loses its grip on the table and falls flat”.The coin spins longitudinally, transversally and even a slight alteration in the angle of the
16、 original impulse alters the resultant trajectory.,Speed of Rotation (w) w = Dq/Dt Note similarity to v = Dx/Dt,Angular Acceleration - Measures how angular velocity is changing (a) a = Dw/Dt Note similarity to a = Dv/Dt,7.8. Basic Concepts,Torque Product of Force and Lever Arm Torque = Force X Lever
17、 Arm,Just as unbalanced forces produce acceleration, unbalanced torques produce angular acceleration.,Center of Mass Average position of the mass of an object Newton showed that all of the mass of the object acts as if it is located there.,Stability In order to balance forces and torques, the center
18、 of mass must always be along the vertical line through the base of support.,7.8. Basic Concepts,Centripetal Force Any force that causes an object to move in a circle. Centripetal forces can be written in different ways:,Fcp = macp Fcp= mv2/r Fcp= mrw2,Centrifugal force Fictitious center fleeing for
19、ce Felt by object in an accelerated reference frame,7.8. Basic Concepts,Angular Momentum L = (rotational inertia) X (angular velocity) L = Iw Compare to linear momentum: p = mv,7.8. Basic Concepts,Linear Momentum and Force Angular Momentum and TorqueLinear SF = Dp/Dt Impulse Dp = SF Dt Rotational St
20、 = DL/ Dt Rotational Impulse DL = St Dt,7.8. Basic Concepts,Conservation of MomentumLinear If SF = 0, then p is constant.Angular If St = 0, then L is constant.,7.8. Basic Concepts,Moment of Inertia,Property of an object that resists changes in rotation For linear motion mass was a measure of inertia
21、 For rotational motion Moment of Inertia (I) is the measure of rotational Inertia Depends on: Mass of the Object Axis of Rotation Distribution of Mass in the Object,7.8. Basic Concepts,7.9. Sources,http:/ http:/physicsweb.org/article/news/4/4/12 http:/hyperphysics.phyastr.gsu.edu/hbase/torcon.html h
22、ttp:/ is a binary that generates a torque It is obvious to affirm that the smaller the , the bigger the torque. As a consequence, the precession movement will be more visible at the end.,Video,It was noticed that at the end of the trajectory, the linear velocity is smaller and F is higher, because:,
23、If v gets smaller, F gets higher and R gets smaller,Video,It should be observed that the coin rotation velocity is reduced simultaneously to the precession velocity when this velocity is higher than the rotation velocity. If the precession movement gets higher, it was obtained that the speed of the
24、point in contact with the floor gets higher and, as a consequence the sound frequency gets higher. Experimentally, it was obtained a 300Hz frequency.,7.4. Results,7.4.4. Appendix: What would happen if the experiment was performed in an non-gravitational environment?,If the experiment was performed i
25、n an environment without any resistive and gravitational forces, there would be no spendthrift forces (Fat=.N). Since the attrition force is responsible for the reduction of the linear speed and the centripetal acceleration, the coin wouldnt make curves and wouldnt have its speed v modified. Therefo
26、re, it is as if the coin remained on the 1st stage of the resulting motion.,7.5. Gyroscope effect,It is a device that: Has almost no dissipative forces Maintain the angular momentum If a force is applied up on the gyroscope, it presents resistance in changing direction A modern version to the top Us
27、ed in planes and spaceships to keep then in courseSee live presentation,7.3. Procedure,7.3.2. calculating the relation between the masses of the coins of R$0,50 and R$0,05.To calculate the relation between the coin masses, a ruler of 30cm was supported on a high surface and a coin was placed in the
28、tip of the ruler. Along with the coin, the ruler was gradually being dislocated out of the high surface until this system fell. The following values had been written down:11,5 cm for the R$0,50 coin 13,5 cm for the R$0,05 coin.,7.3. Procedure,Knowing that the torque can be calculated by M=F.d: (i) M
29、50=m50.g.11,5 = F.3,5 , in which F is the rulers weight (ii) M05=m05.g.13,5 = F.1,5 , in which F is the rulers weight(i)/(ii): M50 = m50.g = 13,5 . 3,5 M05 m05.g = 11,5 . 1,5Then, m50/m05 = 2,739,7.3. Procedure,7.3.3. Verification of the trajectory of the coin on different floorsThe coin motion afte
30、r a slight impulse was observed on different surfaces (wooden floor and sandpaper). It was clear that the trajectory on the sandpaper, that has higher attrition coefficient, was more curved yet faster than the coin behaviour on the wooden surface.,7.3. Procedure,7.3.4. Trajectory filmingThrough a video camera, the described trajectory was filmed on various angles. See tape 1.,
