Spring mass System in Simple Harmonic Motion

College of Science and Computer Science

Physical Sciences Department

Physics 2

Experiment #1

SPRING MASS SYSTEM IN SIMPLE HARMONIC MOTION

(HOOKE’S LAW)

1. Abstract

Hooke’s Law is a principle that states that the force needed to extend or compress a spring by some distance is proportional to that distance. In this experiment, we aim to study the elastic properties of the spring and the force constant of the spring. For a Hooke’s law restoring force, the relationship between the force and the displacement is given by F = -kx where k is called the force (spring) constant. Application of such a force to a mass m yields F = -kx = ma, which is the mathematical statement of the condition for simple harmonic motion. When x = 0 the mass is at the center or equilibrium position. In this experiment we will investigate the elastic behavior on a spiral spring in even greater detail as we focus on how a variety of quantities change over the given time. Such quantities will include distance and tine that will undergo for 5 trials for the results of the experiment to be more accurate. The resulting values of 32.2 N/m, and 21.8 N/m for the force constant, respectively for the two tables, show that the normal spring functions according to Hooke’s Law. The two spiral spring was then used to determine the value of theoretical value and experimental value of the Hooke’s law apparatus by letting the spring’s restoring force drags a heavy mass for 50 vibration per trial. This experiment enables us to compute for the force constant of the spring, frequency, angular frequency, and the period of oscillations of the spring.

2. Actual Materials Used

  • Hooke’s law apparatus (spiral spring, scale, stand)
  • Weight hanger
  • Ruler
  • Stopwatch
  • Set of weights
  • Triple beam balance
  • Paper clip 

 

3. Data and Results 

1st table:

Table 1: Displacements and Times for 50 Vibrations per trial for each load

Load (g) Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average
d1 t1 d2 t2 d3 t3 d4 t4 d5 t5 dave tave
100 0.028 19.07 0.027 18.21 0.028 20.04 0.028 19.17 2.027 19.35 0.0276 19.17
200 0.059 25.19 0.058 27.01 0.059 26.48 0.059 26.31 0.058 26.19 0.0586 26.24
300 0.089 31.41 0.087 31.36 0.088 32.14 0.088 32.06 0.088 31.92 0.088 31.78
400 0.123 26.61 0.124 36.78 0.122 37.27 0.123 37.12 0.123 36.82 0.123 36.92
500 0.152 40.54 0.151 40.64 0.153 41.10 0.152 40.82 0.152 41.07 0.152 40.83

 

Table 2: Experimental Value, Theoretical Value, % Error for Period, Frequency and Angular Frequency

Mass (g) (load) Average Displacement (m) Average Time for 50 Vibrations (sec) Period, T (sec) Frequency, F (Hz) Angular Frequency, ω (rad/sec)
EV TV % Error EV TV % Error EV TV % Error
100 0.028 19.17 0.38 0.35 8.57 2.61 2.86 8.74 16.40 17.94 8.58
200 0.059 26.24 0.52 0.50 4.00 1.91 2.00 4.50 12.00 12.67 5.29
300 0.088 31.78 0.64 0.61 4.91 1.57 1.64 4.27 9.86 10.36 9.65
400 0.123 36.92 0.74 0.70 5.71 1.35 1.43 5.59 8.48 8.97 5.46
`500 0.152 40.83 0.82 0.78 5.13 1.22 1.28 4.67 7.67 8.02 4.36
Force Constant, k= 32.2 N/m
Mass of Spring, ms= 0.007kg Mass of weight hanger, mwh= 0.175kg

Mass of paperclip, mp=0.0015kg

 

 

2nd table:

Table 1: Displacements and Times for 5 Vibrations per trial for each load

Load (g) Trial 1 Trial 2 Trial3 Trial 4 Trial 5 Average
d1 t1 d2 t2 d3 t3 d4 t4 d5 t5 dave tave
100 0.045 22.52 0.041 23.10 0.046 22.96 0.041 22.82 0.042 23.07 0.0424 22.89
200 0.085 30.96 0.086 31.10 0.085 30.85 0.084 31.07 0.087 30.82 0.0854 30.96
300 0.135 38.69 0.137 38.84 0.132 39.00 0.134 38.96 0.136 38.71 0.1348 38.84
400 0.176 43.82 0.178 44.07 0.176 43.72 0.175 44.15 0.172 44.06 0.1754 43.95
500 0.234 48.56 0.236 48.79 0.235 49.02 0.234 48.67 0.237 49.14 0.2352 48.84

 

Table 2: Experimental Value, Theoretical Value, % Error for Period, Frequency and Angular Frequency

Mass (g) (load) Average Displacement (m) Average Time for 50 Vibrations (sec) Period, T (sec) Frequency, F (Hz) Angular Frequency, ω (rad/sec)
EV TV % Error EV TV % Error EV TV % Error
100 0.042 22.89 0.46 0.42 9.52 2.18 2.38 8.40 13.70 14.76 7.18
200 0.085 30.96 0.62 0.60 3.33 1.61 1.67 3.59 10.12 10.44 3.07
300 0.135 38.34 0.76 0.74 2.70 1.31 1.35 2.96 8.23 8.53 3.52
400 1.175 43.95 0.88 0.85 3.53 1.13 1.18 4.24 7.10 7.38 3.79
`500 0.235 48.84 0.98 0.95 3.16 1.02 1.05 2.86 6.41 6.60 2.88
Force Constant, k= 21.8N/m
Mass of Spring, ms= 0.008kg Mass of weight hanger, mwh= 0.175kg

Mass of paperclip, mp=0.0015kg

 

4. Interpretation of Data

In the table 1 for the displacement and time for 50 vibrations per trial for each load. We have done 5 trials for each displacement and time because multiple trials allow us to see whether the results of each of the trials as a whole, show consistency. Consistent findings will reinforce the value of our conclusions. As you can see in the table as the mass increases the displacement and the time also increases. Mass does not affect the displacement directly, but it does affect how fast an object can change its speed by accelerating or decelerating. Objects with larger masses require more time for acceleration or deceleration compared to lighter objects because the greater the mass, the greater the force of gravity that is acting on the object. For any given force, objects with larger masses take longer time to change their speeds than objects with lower masses.

In the table 2 for the Experimental and Theoretical Value, % Error, and the Frequency and Angular Frequency. As you can see the table we have varying amounts for the frequency and period. The quantity frequency is often confused with the quantity period. Frequency and period are distinctly different, yet related, quantities. Frequency refers to how often something happens. Period refers to the time it takes something to happen. Frequency is a rate quantity while Period is a time quantity. You can see in the table as the mass increases the period also increased. This is because of Newton’s Second Law, F = ma. If the mass increases, and everything else remains the same, Newton’s Second Law tells us the acceleration must decrease. The mass will move more slowly. If it moves more slowly, it will require more time to go through one period so the period will increase.

In the table you can see that we have varying amounts of % error in our period and frequency. It means that even if the mass is high or low it will still not affect the performance of the experiment as a whole. This percentage error calculation will help us to evaluate the relevance of our results. It is helpful to know by what percent our experimental values differ to some established value. In most cases, a percent error or difference of less than 10% will be acceptable. As you can see in the table the differences of percent error in our table is less than 10% it means even if are values differ as the mass increases it is still acceptable and the purpose of the experiment still has been accomplished.

Factors affecting our results:

  1. Systematic errors: These are errors which affect all measurements alike, and which can be traced to an imperfectly made instrument or to the personal technique and bias of the observer. In our experiment the weight hanger is not flat and it is bent. So it may affect the precision of the equipment and it can cause varying measurements in displacement which can affect the results of our experiment.
  2. Parallax: This error can occur whenever there is some distance between the measuring scale and the indicator used to obtain a measurement. In our experiment when we looked at the measuring scale hooked in the Hooke’s law apparatus each of our members can see different measurements maybe it is because of the different angles when we look at it or sometimes the indicator (the clip) is too far from the measuring scale. This errors can result into different values in the displacement which can affect the results of our experiment.
  3. Personal errors – Carelessness, poor technique, or bias on the part of the experimenter. In our experiment as we count the time for 50 vibrations per trial some us counted too fast and some of us counted to slow this results to different values in the time which can affect the results of our experiment.

If the experiment was to be conducted again, these sources of error would have to be minimized to make the results of the experiment more valid. We can do that by using more precise and accurate equipment, learn to obtain measurements more precise and by being more vigilant in calculating and monitoring the data being collected and used.

5. Conclusion

In conclusion we use Hooke’s Law in spring balances, kitchen scales and other devices where we measure using a spring. A spring is an object that can be deformed by a force and then return to its original shape after the force is removed. Hooke’s Law says that the stretch of a spring is directly proportional to the applied force. In symbols, F = kx, where F is the force, x is the stretch, and k is a constant of proportionality. If Hooke’s Law is correct, then, the graph of force versus stretch will be a straight line. The amount the spring stretches and was plotted against the weight added to the hanger gives a straight line that goes through the origin. This means that the extension of a spring is directly proportional to the stretching force applied to it.

 

6. Guide Questions

a. Give 5 applications of simple harmonic motion of the type spring-mass system and their corresponding functions.

Pendulum

Any suspended object moving from side to side can be considered to be a pendulum. For example chandelier swaying in the breeze. Sometimes it was barely moving and at other times it was swinging in wide arc but one complete swing always took the same amount of time.

Diving Board

Once the diver has left the board, it will oscillate up and down. The board is one example of a very common situation called a cantilever. A cantilever is a rigid structural element, such as a beam or a plate, anchored at only one end to a (usually vertical) support from which it is protruding.

Trampoline (bungee jumping)

The fall from the platform is an acceleration due to gravity but once the elastic material has straightened out. The jumper’s mass is on the end of the spring and so it oscillates vertically

The Torsional Oscillator

Consider a circular disk suspended from a wire fixed to a ceiling. If the disk is rotated, the wire will twist. When the disk is released, the twisted wire exerts a restoring force on the disk, causing it to rotate past its equilibrium point, twisting the wire the other direction, as shown below. This system is called a torsional oscillator.

Object floating and bobbing in water

Any object floating in water can bob up and down. If the cross-section going into the water and the object is approximately constant. Then the oscillation will be a simple harmonic motion (SHM). For example a testube containing a small mass will float in water and bob up and down in the water projecting a simple harmonic motion.

b. A spring has its own force constant. What happens to this force constant if this spring is cut into half? Explain your answer.

The force constant doubles if the spring is cut in half. For example by applying an equal force F to each end while the center remains unmoved. The spring constant is k=F/(2x), where x is the distance each end moves. Effectively, the spring compresses by a distance 2x overall. But now look at only the left half of the spring. The force F caused a compression of x only. The new spring constant for the half spring is therefore F/x=2k. So in general the spring constant of a spring is inversely proportional to the number of coils in the spring.

c. A small block of mass 200g is used as a harmonic oscillator attached to one end of an ideal spring of which force constant is unknown. If the oscillator is found to have a period of 0.30s, find the force constant of the spring.

 

7. References

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