Simple Harmonic Motion Introduction: Simple harmonic motion, as shown in Figure 1, is any motion that has a restorative force and forces the object to oscillates indefinitely in the absent of frictional forces. For the case of a mass attached to a spring the restorative force is given by Hooke's law E = kx. m_ ot trarn vl el 1= 47 Pma_lm_bm_ mr=T "'.'IT,'TI'E'U?.'I" Figure 1: l]lustrano-n uf simple harmonic motion for a spring-mass combination at different times, t, of its motion after being released from the x = 4 position. _ Another example of simple harmonic motion is the motion of pendulum where the restorative force is given by F=--mgsinf = mygh mg _ L x_. where the small angle approximation was used to transform the restorative force from a trigonometric function to a linear function in similarity to Hooke's law. The equation of motion for an object - oscillating in simple harmonic motion is g y = Asin{awt + ), Figure 2: lustration of simple harmonic where y is the distance from the equilibrium motion for a pendulum of mass m, position at any time , 4 is the amplitude, or length L, and displacement x. the maximum displacement of the object, and g1is the initial phase angle The oscillatory nature of the simple harmonic motion leads us to compare to circular motion. Therefore the angular frequency can be related to the period of oscillation and the equation of motion can also be expressed as = Asin(2rft) = Asin (ETEJ Laboratory #01: Simple Harmonic Motion where fis the frequency of oscillation and T is the period of oscillation. Further inspection allows us to express the period of oscillation, the frequency of oscillation and the angular speed as function of the spring constant and the mass attached to it: ||m 1 |k k T=2n| = [ = [| Nk f 2 m " m respectively. However it is important to point that springs hardly ever behave as ideal springs. Therefore, in the previous relationships for period, frequency and angular speed we need to use the effective mass of the system, m.y. given by Moy = M panygur + E M gpring Similar expressions can be obtained to describe the oscillation motion of a simple pendulum. L 1 (g T r=2x |- =i _ "J; f =22l ST Equipment Pasco 850 Universal Interface (UL-3000) DMotion sensor 11 (CI-6742) * Spring # Mass hanger with an additional 300 g masses Determination of Spring Constant k Oscillation 1. 2 3. e Measure the mass of the spring. mass hanger with a 500 g mass. Write results on the provided space and calculate the effective mass. Hang the mass-spring combination from the bar on the metal stand. Pull the mass-spring combination down slightly and release it to create small oscillations. Measure the time required for 5 oscillations. Write results in Table 1. a. To obtain consistent measurement make sure that every time that you pull the mass-spring combination you do it by the same distance, for example by 5 em. For this use the meter-stick as a reference. Repeat the previous measurement six times and calculate the average time. . Calculate the period for the oscillating of the mass-spring system for the average time, i T=, . Calculate the spring constant of the spring using your knowledge of the mass-spring mass and period of oscillation. a. Make sure that vou use the effective mass in this caleulation. b. Calculate percent difference comparing with the value obtained in the previous part. Table | T= T provided space and calculate the effective Hang the mass-spring combination from the bar on the me Pull the mass-spring combination down slightly and release it to create Measure the time required for 5 oscillations. Write results in Table 1. a. To obtain consistent measurement make sure that every time that you pull the mass-spring combination you do it by the same distance, for example by 5 cm. For this use the meter-stick as a reference. su Repeat the previous measurement six times and calculate the average time. Calculate the period for the oscillating of the mass-spring system for the average time, of er T = = 6. Calculate the spring constant of the spring using your knowledge of the mass-spring mass and period of oscillation. you Make sure that you use the effective mass in this calculation. Obje Calculate percent difference comparing with the value obtained in the previous part. mhanger = 49.8 9 Plastic C mspring = 4. 2 J meffective = Alumini Table 1 Cylinde Trial t (sec) 2 _ 4 T = Brass N 2. 6 k = 2 . 7 Percent difference = in 3.15 2. 7 Average 15.8- 182