Pendulum and Springs Lab Report

From the Free body Diagrams, we are able to apply Equation (1) and derive the following workup shown in (2) which explains both the forces that are being applied in the X and Y direction in the pendulum to get the total forces being applied to the pendulum as a whole. Σ Fx = m a X = Tsinθ Σ Fy = m a y = Tcosθ -mg (2) ΣT = Iα = m L 2 α = mgsinθ Using equations 1 and 2 we are able to derive equation 3 where T is the period of oscillation. This is measures as the time the pendulum takes to complete a full motion. T = 2 π ω (3) To solve for T we need to first solve for ω which is the angular frequency and the equation for that is shown in equation 4. Where L is the length of the pendulum. ω = √ g L (4) Substituting equation (4) into equation (3) we are able to get the final equation for the period of oscillation that is shown in equation (5) T = 2 π √ g L (5) In the second part of lab, we are measuring the spring constant (x) the equation for this originates from newtons second law as well shown in equation 1. Therefore, we are able to use a free body diagram to solve for the spring constant. Where k is the spring constant and x is the displacement from equilibrium. kx Mg Using equation 1 and the free body diagram we are able to derive equation (6) ma =− kx (6) The angular frequency of the mass and the spring can be expressed with equation (7) k is the spring constant that we are looking to solve, and m is the mass that is being added to the spring.