actual experiment, the actual displacement of struck ball x is al ways smaller tha the distance x. Similar with the theoretical expression of initial height ho, you can easily write down the theoretical expression of h' which corresponds to the The lost energy in this trial is: AE= 7) In order to compensate the energy loss in previous trial, we should increase the initial gravitational potential energy of strike ball, in other words, the start height of the strike ball should be raised a little: 4H= 8)Procedure 7)may be repeated for several times until your " score "is better than 9th circle(inclusive)
5 actual experiment, the actual displacement of struck ball x’ is always smaller than the distance x. Similar with the theoretical expression of initial height h0, you can easily write down the theoretical expression of h’ which corresponds to the displacement x’: h’= . The lost energy in this trial is: ∆E’= . 7) In order to compensate the energy loss in previous trial, we should increase the initial gravitational potential energy of strike ball, in other words, the start height of the strike ball should be raised a little: H= . 8) Procedure 7) may be repeated for several times until your “score” is better than 9 th circle (inclusive)
2. Data Record 1)Do some adjustments in order to make sure your strike ball and struck ball had a head-on collision 2) Try the experiment once to verify your calculation 3)If the center of the target-sheet is not hit, analyze the possible reason(s)and make a new calculation. Readjust the device if necessary, especially the start height of the strike ball. (Write down the details of your observation, alculation and adjustment. ) Try again to verify the new calculation Data table No. 1: (Pay attention to the significant figures of your values. D/cm m/g m/g y/cm x/cm ho( Calculated)/em Data table no. 2: Shot position Possible reasons for the AEJJ Ho/cm No. Score 4H,/em (x', z")/cm ' miss-hit (average) H1= Shot position Possible reasons for the AEJ N Score 4H2/cm Ho+AHI x,4)/cm smiss-hit' avera Shot position Possible reasons for the AEvJ No. Scor 4H3/cm H1+4H2 (x', z)/cm smiss-hit' (average) Total energy loss: Ae= zis the distance of the shot position to the middle line of the target paper. It should a smal value 6
6 2. Data Record 1) Do some adjustments in order to make sure your strike ball and struck ball had a head-on collision. 2) Try the experiment once to verify your calculation. 3) If the center of the target-sheet is not hit, analyze the possible reason(s) and make a new calculation. Readjust the device if necessary, especially the start height of the strike ball. (Write down the details of your observation, calculation and adjustment.) Try again to verify the new calculation. Data table No. 1: (Pay attention to the significant figures of your values.) D/cm m1/g m2/g y/cm x/cm h0(Calculated)/cm Data table No. 2： z' is the distance of the shot position to the middle line of the target paper. It should be a small value. H0/cm No. Score Shot position (x', z')/cm Possible reasons for the "miss-hit" E1/J (average) H1 /cm 1 2 3 H1 = H0+H1 No. Score Shot position (x', z')/cm Possible reasons for the "miss-hit" E2/J (average) H2 /cm 1 2 3 H2 = H1+H2 No. Score Shot position (x', z')/cm Possible reasons for the "miss-hit" E3/J (average) H3 /cm 1 2 3 Total energy loss: E=
Lab 2 Torsional pendulum Goal This experiment is designed for a review of the rotation of rigid body Related topics Rotational motion, Oscillatory motion, Elasticity Introduction A torsional pendulum, or torsional oscillator, consists usually of a disk-like mass suspended from a thin rod or wire(see Fig. 1). When the mass is twisted about the axis of the wire, the wire exerts a torque on the mass, tending to rotate it back to its original position. If twisted and released the mass will oscillate back and forth, executing simple harmonic motion. This is the angular version of the bouncing mass hanging from a spring. This gives us an idea of moment of inertia We will measure the moment of inertia of several different shaped objects. As comparison, these moment of inertia can also be calculated theoretically. We can also verify the parallel axis theorem Given that the moment of inertia of one object is known, we can determine the torsional constant Fig. I Schematic diagram of a torsional pendulum This experiment is based on the torsional simple harmonic oscillation with the analogue of displacement replaced by angular displacement e, force by torque M, and the spring constant by torsional constant K. For a given small twist 6(sufficiently small), the experienced reaction is given by M=-K8 (1) This is just like the Hooke's law for the springs. If a mass with moment of inertia / is attached to the rod, the torque will give the mass an angular acceleration a according to m=a=/ get the following d-e K 6 Hence on solving this second order differential equation we get 8=Acos(t+o)
7 Lab 2 Torsional pendulum Goal This experiment is designed for a review of the rotation of rigid body Related topics Rotational motion, Oscillatory motion, Elasticity Introduction A torsional pendulum, or torsional oscillator, consists usually of a disk-like mass suspended from a thin rod or wire (see Fig. 1). When the mass is twisted about the axis of the wire, the wire exerts a torque on the mass, tending to rotate it back to its original position. If twisted and released, the mass will oscillate back and forth, executing simple harmonic motion. This is the angular version of the bouncing mass hanging from a spring. This gives us an idea of moment of inertia. We will measure the moment of inertia of several different shaped objects. As comparison, these moment of inertia can also be calculated theoretically. We can also verify the parallel axis theorem. Given that the moment of inertia of one object is known, we can determine the torsional constant K. Fig. 1 Schematic diagram of a torsional pendulum This experiment is based on the torsional simple harmonic oscillation with the analogue of displacement replaced by angular displacement , force by torque M, and the spring constant by torsional constant K. For a given small twist (sufficiently small), the experienced reaction is given by M K (1) This is just like the Hooke’s law for the springs. If a mass with moment of inertia I is attached to the rod, the torque will give the mass an angular acceleration according to 2 2 d M I I dt Then we get the following equation: 2 2 d θ K dt I (2) Hence on solving this second order differential equation we get: A t cos( ) (3)
T=2丌 Where, is the angular velocity of simple harmonic oscillation, and T is the period. So, if period Tand the K are known, the inertia of the rotated object can be expressed as K 4 The additive property of inertia: The moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems(all taken about the same axis) =∑l(=123 Experiment device LED late spindle chopping bardetector aIrscrew sprIn power display ase 5 cycles reset Fig 2 Schematic diagram of the experiment device Procedure Familiarize yourself with the operation of the device. Adjust the device carefully so that it is ready for the measurement(i.e. the base is placed horizontally Determine the torsional constant K of the spiral spring with a plastic cylinder. Employ the theoretically calculated moment of inertia of the plastic cylinder as a known value Measure the moment of inertia of differently shaped objects. Compare the measured results with the theoretically calculated values Optional Verify the parallel axis theore
8 K I (4) 2 T I K (5) Where, is the angular velocity of simple harmonic oscillation, and T is the period. So, if the period T and the K are known, the inertia of the rotated object can be expressed as: 2 2 4 K I T (6) The additive property of inertia: The moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis): ( 1,2,3...) j j I I j (7) Experiment device Fig. 2 Schematic diagram of the experiment device Procedure Familiarize yourself with the operation of the device. Adjust the device carefully so that it is ready for the measurement (i.e. the base is placed horizontally.). Determine the torsional constant K of the spiral spring with a plastic cylinder. Employ the theoretically calculated moment of inertia of the plastic cylinder as a known value. Measure the moment of inertia of differently shaped objects. Compare the measured results with the theoretically calculated values. Optional Verify the parallel axis theorem
Questions 1. On which factors is the moment of inertia dependent 2. What are the causes that may bring error to our measurement? References 1.沈元华,陆申龙基础物理实验( Fundamental Physics Laboratory)高等教育出版社北 京2004pp.100-103 2.RavitejUppuTorsionalPendulumhttp://www.cmi.ac.in/-ravitej/lab/4-torpen.pdf Appendix: l1 Uncertainty of Plastic cylinder: D&u(Ipr) u(m) (D) D u(D)=√l2(D)+n2(D) l(m)=√x2(m)+2(m Where, uBI(m)=d=0.1g u(D)= ∑(D-D 0.2 n(n-1) g 0.02 [2 Uncertainty of K l(K)= l(c)|a(72-7) 70 Please note: (2-73)=√(2)3+[(7)&7)=27×(T) Since we measure the time of 5 periods, so for each period, we have T==t and ()=√:()+2( ∑ n(n-1) 0.01 √3√3 S
9 Questions 1. On which factors is the moment of inertia dependent? 2. What are the causes that may bring error to our measurement? References 1. 沈元华，陆申龙 基础物理实验 (Fundamental Physics Laboratory) 高等教育出版社 北 京 2004 pp. 100-103 2. Ravitej Uppu Torsional Pendulum http://www.cmi.ac.in/~ravitej/lab/4-torpen.pdf Appendix: [1] Uncertainty of Plastic cylinder: 1 1 2 2 2 8 PC I mr mD & 2 2 ( ) (D) ( ) 2 PC PC u m u u I I m D Where, 1 2 2 2 1 2 ( ) ( ) ( ) ( ) 0.1 0.2 ( ) 3 3 B B B B u m u m u m u m d g a u m g & 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( 1) 0.02 ( ) 3 3 A B i A B u D u D u D D D u D n n a u D mm . [2] Uncertainty of K. 2 2 2 2 0 2 2 0 ( ) ( ) ( ) PC PC u I u T T u K K I T T . Please note: 2 2 2 2 2 2 0 0 u T T u T u T ( ) [ ( )] [ ( )] & 2 u T T u T ( ) 2 ( ) . Since we measure the time of 5 periods, so for each period, we have 1 5 T t and 1 ( ) ( ) 5 u T u t . Where, 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( 1) 0.01 ( ) 3 3 A B i A B u t u t u t t t u t n n a u t s