4-Dec-2016: Physical Pendulum Lab

Apperetus: A narrow metal ring and a semimcircular plate.

Model: τ=αI

Purpose: Derive expressions for the period of various physical pendulums. Verify your predicted periods by experiment.

Process: Part A:
    First, we detemine the outer radius and inner radius, router=0.1465 m,  rinner=0.1372 m. Because the ring is thin enough, we assume that the point of pivot is exactly half way between the inner and outer radii and that the notch, which is cut out at the top of the ring, does not impact how uniform the mass distribution is. We hang the ring and set up the sensor. We raise the ring with a small angle, release it, and use the LoggerPro to collect date. By the data, we can get the period.

    We get the exprimental period, T=1.06977188 s.
    Then, we need to calculate out the theoretical period:
        We assume the radius of the ring is the average of the inner radius and outer radius.
                                                      R=router+rinner=0.14185 m
        We already calculate the equation of the period in the notebook
                                                      T=2π(2R/g)^0.5
        We get the theoretical period:
                                                      T=1.0690 s
        We compare them:
                                                      Experimental T=1.06977188 s
                                                      Theoretical T=1.0690 s
                                                      0.7% off

    Part B:
    Now, we have a semicircular disk of radius R, R=0.1445 m.
    

    We hang the semicirular disk as the picture. We raise the semicirular with a small angle, release it, and use the LoggerPro to collect date. By the data, we can get the period.

    We get the exprimental period, T=0.83722056 s.
    Then, we need to calculate out the theoretical period:
                                                            τ=αI
                                                            sinθ*mg*(4/3π)R=α*(1/2)mR^2
                                                            α=(8g/3πR)sinθ
    When the angle is small enoug, we can assume θ≈sinθ.
                                                            α=(8g/3πR)sinθ≈(8g/3πR)θ
                                                            ω=(8g/3πR)^0.5
                                                            T=2π/ω=2π(3πR/8g)^0.5
    We get the theoretical period:
                                                            T=0.82812 s
    We compare them:
                                                      Experimental T=0.83722056 s
                                                      Theoretical T=0.82812 s
                                                      1.3% off

    We hang the semicirular disk as the picture. We raise the semicirular with a small angle, release it, and use the LoggerPro to collect date. By the data, we can get the period.

    We get the exprimental period, T=0.81718602 s.
    Then, we need to calculate out the theoretical period:
                                                            τ=αI
                                                            sinθ*mg*(1-4/3π)R=α*(1/2)mR^2
                                                            α=[(6π-8)g/(9π-16)R]sinθ
    When the angle is small enoug, we can assume θ≈sinθ.
                                                            α=[(6π-8)g/(9π-16)R]sinθ≈[(6π-8)g/(9π-16)R]θ
                                                            ω=[(6π-8)g/(9π-16)R]^0.5
                                                            T=2π/ω=2π[(9π-16)R/(6π-8)g]^0.5
    We get the theoretical period:
                                                            T=0.811509744 s
    We compare them:
                                                      Experimental T=0.81718602 s
                                                      Theoretical T=0.811509744 s
                                                      0.85% off