Physics Simple Harmonic Motion Lab Report PDF

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Simona Khan Brandon Byars Jewel Taylor

11/16/17 Simona Khan

Experiment # 9 Simple Harmonic Motion

Object: The object of this experiment is to study two examples of simple harmonic motion, the vibrating spring and the simple pendulum

Theory: An object undergoes simple harmonic motion when it is subject to a force proportional to its displacement from an equilibrium position. The law governing this motion is known as Hooke's Law and is given by F = - kx, where F is the restoring force and the minus sign indicates that the vibrating force and the displacement, x, are in opposite directions. The proportional constant, k, is the force constant of the spring.

Apparatus: 1) 2) 3) 4)

Spiral spring and rigid support Set of weights Meter stick Quadruple – beam balance

Procedure A: For vibrating spring 1) Weigh and record the mass of the spring 2) Suspend the spring from the rigid support with the smaller diameter attached to the support. 3) Place a meter stick vertically close to the spring with the 100 cm end resting on the table top. Use the bottom end of the spring as a reference point. Record this initial position which corresponds to the no load point. 4) Place a 50 gram mass on the end of the spring and record the elongations of the spring from the reference point. 5) Repeat step 4 with 100, 150, 200, 250 and 300 grams 6) Place a 100 gram mass on the end of the spring, pull the mass downward a few centimeters and release it. Measure the time for 20 complete vibrations. Record in one column the time for 20 vibrations and in an adjacent column the mass used. 7) Repeat step 6 for 150, 200, 250 and 300 grams.

Data: Table 1 – Data for Spring Constant k Load on Spring m (gm) Meter Stick Scale Reading (cm)

Elongation For Each Load (cm)

m + = ms/3

0

57

0

52.67

50

51

6

102.67

100

45.5

11.5

152.67

150

39.5

17.5

202.67

200

32

25

252.67

250

26

31

302.67

300

19.9

37.5

352.67

Table 2 – Data for the Period of the Spring Load on Spring m (gm)

Time for 20 Vibrations (sec)

Period Time for One Vibration (sec)

Period from Equation 3 (sec)

Percent Difference

100

15.98

0.80

0.89

10.04%

150

19.4

0.97

1.02

5.21%

200

22.5

1.13

1.14

1.54%

250

23.71

1.19

1.25

5.20%

300

25.46

1.27

1.35

5.70%

Analysis:

Questions: 1) It is not necessary to add 1/3 of the mass of the spring to the load when plotting the graph to obtain k because it would not change the slope of the graph, only the y-intercept would change. 2) The value for T is 0.99 and the percent error is 7.94%. 3) The two graphs differ because the one including mass of spring is much smoother and uniform. 4) (a) The point in the path that the spring exerts the greatest force on the suspended mass is at the bottom of the oscillation, where acceleration is the greatest midway between being released and the bottom. (b) Velocity is greatest at the same spot as acceleration. 5) The potential energy stored is F=-7.66*.05 6) Small changes do not change because the equation does not reference the amplitude. 7) T= 2pi*sqrt(m/k) so T= 5.5 sec

Part B: Theory – A simple pendulum is the second example of a system that exhibits simple harmonic motion. It consists of a small bob of mass m suspended by a light string of length l and fixed at one end. When released the bob swings to and fro over the same path. For small angles, less than 10 degrees, the restoring force is given by F = -(mg/l) x, where mg is the weight of the bob, l is the length of the pendulum and x is the horizontal displacement of the bob from its equilibrium position.

Apparatus: 1) 2) 3) 4) 5)

Metal sphere (bob) on a long string (The simple pendulum) Rigid support with pendulum clamp Timer Vernier Caliper Rule

Procedure B For the Simple Pendulum: 1) Measure the diameter of the metal sphere with the vernier caliper 2) Clamp the string so that the top of the sphere is about 90 cm below the point of support. Record this distance as the length of the string used. The length of the pendulum will be this distance plus the radius of the metal sphere. 3) Displace the pendulum from its rest position through a very small angle (less than 10 degrees) and release it, being careful not to impart a twisting or circular motion and record the time it takes the pendulum to make 20 vibrations. 4) Repeat steps 3 and 4 of Procedure B, making the length of the pendulum successively 80, 70, 60, 40 and 30 cm. 5) With a pendulum length of 30 cm, start the pendulum vibrating through a large arc of 40 degrees for 20 vibrations and record the result on Table 4.

Analysis:

Data:

Diameter of sphere: 2.5 cm

Radius of sphere: 1.25 cm

Length of String (cm)

Length of Number of Pendulum (cm) Vibrations

Time for 20 Period (sec) Vibrations (sec)

Square of Period (sec^2)

90

92.5

20

36.73

1.84

3.37

80

82.5

20

34.81

1.74

3.03

70

72.5

20

32.8

1.64

2.69

60

62.5

20

30.56

1.53

2.34

50

52.5

20

28.69

1.43

2.06

40

42.5

20

25.95

1.3

1.68

30

32.5

20

23.36

1.17

1.36

Number of Vibrations

Time for 20 Vibrations (sec)

Period (sec)

20

21.27

1.06

Questions: 1) 2) 3) 4) 5) 6) 7) 8)

The period remains the same. It would not change. Acceleration due to gravity doesn’t change depending on mass. It would also not change. Mass is not part of the period equation. Because the sine of a small angle is close to zero and this reduces the equation to a simple approximation. We need the mass of the bob and the length of the pendulum. Largest velocity at the bottom of the swing. Largest P.E. at the top. Force is greatest at the bottom. Acceleration is greatest at the top. 9.8/1.66=x/1 so 5.9 seconds...