Simple Harmonic Motion Introduction: Simple harmonic motion, as shown in Figure 1, is any motion that...

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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 Fs = -kx. +000000 000000 x-0 --+ (a) Equilibrium Ib) 0 Just before release F-0 000000 -T 000000 000000 - M-T Figure 1: Illustration of simple harmonic motion for a spring-mass combination at different times, t, of its motion after being released from the x = 4 position. L mg sin e mg mg cos Figure 2: Illustration of simple harmonic motion for a pendulum of mass m, length L, and displacement x. Another example of simple harmonic motion is the motion of pendulum where the restorative force is given by F = -mg sin 0 =-mg0 mg 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 y = A sin(wt + ), where y is the distance from the equilibrium position at any time t, A is the amplitude, or the maximum displacement of the object, and is 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 y = A sin(2ft) = A sin Laboratory #01: Simple Harmonic Motion where fis the frequency of oscillation and 7 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: T = 2 2 m W = 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, me, given by meff=mhanger + spring Similar expressions can be obtained to describe the oscillation motion of a simple pendulum. T = 2 f= 1 g 2 L' Equipment Pasco 850 Universal Interface (UI-5000) Motion sensor II (CI-6742) Spring Mass hanger with an additional 500 g masses W = Laboratory #01: Simple Harmonic Motion where fis the frequency of oscillation and 7 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: T = 2 2 m W = 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, me, given by meff=mhanger + spring Similar expressions can be obtained to describe the oscillation motion of a simple pendulum. T = 2 f= 1 g 2 L' Equipment Pasco 850 Universal Interface (UI-5000) Motion sensor II (CI-6742) Spring Mass hanger with an additional 500 g masses W = III. Laboratory #01: Simple Harmonic Motion Determination of g- Simple Pendulum 1. Measure the mass of the metal plumb bob. 2. Construct a simple pendulum 100.0 cm in length using the plumb bob. a. L is measured from the center of mass of the plumb bob. 3. Move the pendulum from equilibrium to about 20 and release it. Using a stopwatch measure the time required for 5 oscillations. Record measurements in Table 2. 4. Repeat the previous measurement five more times. 5. Shorten L in increments of 20.0 cm and measure the corresponding / required for ten oscillations. Mpendulum Table 2: Trial L= 1.0 m L = 0.8 m 1 7.56 7.4 L = 0.6 m 2.73 L=0.4 m L = 0.2 m sity 2 6.76 6.93 7.28 3 7.33 7.12 6.95 4 6.63 6.68 6.44 5 6.13 6.18 6.08 6. Calculate the average time for ten oscillations for each length. Write results in Table 3. 7. From the average oscillation time determine the period for each length of the pendulum and square period. Record results in Table 3. Table 3 L (m) Average t for 5 oscillations (sec) T (sec) T (sec) 1.0 0.8 plum 0.6 0.4 0.2 8. Create a graph of T2 vs. L. a. Apply a linear trendline to the data to determine its slope. From the slope determine the value of g. and calculate percent error. slope = g= b. Should this linear plot have a y-intercept or not? Why? 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 Fs = -kx. +000000 000000 x-0 --+ (a) Equilibrium Ib) 0 Just before release F-0 000000 -T 000000 000000 - M-T Figure 1: Illustration of simple harmonic motion for a spring-mass combination at different times, t, of its motion after being released from the x = 4 position. L mg sin e mg mg cos Figure 2: Illustration of simple harmonic motion for a pendulum of mass m, length L, and displacement x. Another example of simple harmonic motion is the motion of pendulum where the restorative force is given by F = -mg sin 0 =-mg0 mg 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 y = A sin(wt + ), where y is the distance from the equilibrium position at any time t, A is the amplitude, or the maximum displacement of the object, and is 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 y = A sin(2ft) = A sin Laboratory #01: Simple Harmonic Motion where fis the frequency of oscillation and 7 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: T = 2 2 m W = 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, me, given by meff=mhanger + spring Similar expressions can be obtained to describe the oscillation motion of a simple pendulum. T = 2 f= 1 g 2 L' Equipment Pasco 850 Universal Interface (UI-5000) Motion sensor II (CI-6742) Spring Mass hanger with an additional 500 g masses W = Laboratory #01: Simple Harmonic Motion where fis the frequency of oscillation and 7 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: T = 2 2 m W = 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, me, given by meff=mhanger + spring Similar expressions can be obtained to describe the oscillation motion of a simple pendulum. T = 2 f= 1 g 2 L' Equipment Pasco 850 Universal Interface (UI-5000) Motion sensor II (CI-6742) Spring Mass hanger with an additional 500 g masses W = III. Laboratory #01: Simple Harmonic Motion Determination of g- Simple Pendulum 1. Measure the mass of the metal plumb bob. 2. Construct a simple pendulum 100.0 cm in length using the plumb bob. a. L is measured from the center of mass of the plumb bob. 3. Move the pendulum from equilibrium to about 20 and release it. Using a stopwatch measure the time required for 5 oscillations. Record measurements in Table 2. 4. Repeat the previous measurement five more times. 5. Shorten L in increments of 20.0 cm and measure the corresponding / required for ten oscillations. Mpendulum Table 2: Trial L= 1.0 m L = 0.8 m 1 7.56 7.4 L = 0.6 m 2.73 L=0.4 m L = 0.2 m sity 2 6.76 6.93 7.28 3 7.33 7.12 6.95 4 6.63 6.68 6.44 5 6.13 6.18 6.08 6. Calculate the average time for ten oscillations for each length. Write results in Table 3. 7. From the average oscillation time determine the period for each length of the pendulum and square period. Record results in Table 3. Table 3 L (m) Average t for 5 oscillations (sec) T (sec) T (sec) 1.0 0.8 plum 0.6 0.4 0.2 8. Create a graph of T2 vs. L. a. Apply a linear trendline to the data to determine its slope. From the slope determine the value of g. and calculate percent error. slope = g= b. Should this linear plot have a y-intercept or not? Why?