Question

A pendulum consists of a string of length L and a mass m hung at one end and the mass oscillates along a circular arc. Part a) Familiarize yourself with the derivation of omega = Squareroot g/L to hold. i) Explain succinctly how the angular frequency of oscillation omega = Squareroot g/L comes about from Newton's Law, where g is the gravitational acceleration. ii) One assumption required is the small angle approximation: sin theta = theta and cos theta = 1 - theta^2/2. State other assumptions for this equation omega = Squareroot g/L to hold. Part b) Show that energy is conserved at all times: Representing the amplitude A of the oscillation by the are length L theta_A, where theta_A is the maximum angle made by the pendulum swing, we may write x(t) = L theta_A cos(omega t). The angle of the pendulum as a function of time is then given by theta(t) = x(t)/L. i) Find the expression for the velocity upsilon(t). ii) Based on theta(t) and upsilon(t), express the gravitational potential and kinetic energies as a function of time. iii) Using small angle approximation and omega = Squareroot g/L, show that the total energy is constant in time. Part c) Measuring gravitational acceleration: Construct a pendulum. Measure its length and time ten cycles of oscillation. Determine the value of g from this experiment.

Answer 1

Question 3 A pendulum consists of a string of length L and a mass m hung at one end and the mass oscillates along a circular Part a) Familiarize yourself with the derivation of w = Vg/L: Shi is m em md x. i Explain suscrictly how the angular fre quency ofcescillation w g is the gravitational acceleration. む。mes akou from Newton's law, where V ii) One Bssumption required is the small angle appro imation; sinθ s θ and case ~ 1 assumptions for this equationwvg/L to hold. g2/2. State other Part b) Show that energy is conserved at all times Representing the amplitude A of the oscination by the arc length Lea, where θ 1 is the maximum angle made by the pendulin swing, we may write z(t) = LBao08(wt). The angle of the pendulum as a function otme is then given by 0(t) /L i) Find the expression for the velocity v(t). ii) Based on θ(t) and y(t), express the gravitational potential and kinetic energies as a function of time. iii) Using small angle approximation and ω-Vg/L, show that the total energy is constant in time. 1-29 Part c) Construct a pendulum. Measure its length and time ten cycles of oscillation. Determine the value of g from this Nuuo