Pendulum
Pendulum period in seconds T ≈ 2π√(L/g) or, rearranging: g ≈ 4π²L/T² L ≈ T²g/4π² L is length of pendulum in meters g is gravitational acceleration (on earth, nominal of 9.8 m/s²) Length for ½ second = 0.062 m Length for 1 second = 0.248 m Length for 2 second = 0.993 m Length for 4 second = 3.97 m Uniform beam pendulum center of oscillation is 2/3 of the length of the uniform beam L from the pivoted end. T ≈ 2π√((2L/3g) Compound pendulum Any swinging rigid body free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum. The appropriate equivalent length L for calculating the period of any such pendulum is the distance from the pivot to the center of oscillation. This point is located under the center of mass at a distance from the pivot traditionally called the radius of oscillation, which depends on the mass distribution of the pendulum. If most of the mass is concentrated in a relatively small bob compared to the pendulum length, the center of oscillation is close to the center of mass. The radius of oscillation or equivalent length L of any physical pendulum can be shown to be: L = I/mR where I is the moment of inertia of the pendulum about the pivot point, m is the mass of the pendulum, and R is the distance between the pivot point and the center of mass. The period of a compound pendulum is given by T = 2π√(I/mgR) for a uniform rod pivoted at one end, I = mL²/3 T = 2π√(2L/3g) Speed of bob at lowest point O is the pivot point of the pendulum. A is the release point B is the point of lowest travel θ is the deflection angle at release φ is angle ABO α is angle BAC L = OA = OB = length of string OC = OA cos θ h = BC = OA – OC = OA – OA cos θ = OA (1 – cos θ) Now use KE = PE to get the speed ½mV² = mgh V = √(2gh) Pendulum period in sec for larger angle, θ in radians T ≈ [2π√(L/G)] [1 + (1/16)θ² + (11/3071)θ⁴ + (173/737280)θ⁶ + (22931/1321205760)θ⁸ + ... ] T ≈ [2π√(L/G)] [1 + (1/2)²(sin²(θ/2) + ((1·3)/(2·4))²(sin⁴(θ/2) + ((1·3·5)/(2·4·6))²(sin⁶(θ/2) + .. ]Conical Pendulum period T = 2π√(r/(gtanθ)) r is radius of the rotation θ is angle of bob with vertical v = 2πr/T v is velocity Change in simple pendulum period due to length change dT/dL = (1/2)2π√(1/g)√1/L) = π√(1/Lg) dT = (π/√(gL))dL for a 1% change in length, what is time change for 1 day, 86400 seconds? dL/L = 0.01 (ie, 1%) dL = 0.01L dT = π/√(gL) (0.01)L dT = (0.01)(π/√g) L/√L dT = (0.01)(π/√g)√L dT = (0.01)(π√(L/g)) dT = (0.01)(1/2)(2π√(L/g)) dT = (0.005)T dT = (0.005)(86400) = 432 seconds Conical Pendulum where the bob moves in a circle. T = tension in string of length L m = mass of bob θ = angle of string with vertical t = 2π√(Lcosθ/g) Compound Pendulum where the rod is not massless or the bob is extended. T ≈ 2π√(I/mgL) g is gravitational acceleration = 9.8 m/s² I is moment of inertia around the pivot in kg·m² m is the mass of the pendulum in kg L is the distance from the pivot to the center of mass of the pendulum in meters Moment of inertia, I, in kg·m² I = cMR² M is mass (kg), R is radius (meters) c = 1 for a ring or cylinder c = 2/5 solid sphere c = ½ solid cylinder or disk around its center c = 1/12 rod around its center, R = length |
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