Determination of natural frequency (f)

Table 6.2: Result of experiment to find out the natural frequency (f).

no.mass (kg)total mass (kg)Length for 5 oscillation (mm)Period Time, T (s)Frequency (Experimental) , (Hz)Frequency (Theoretical) , (Hz)Percentage Error (%)

101.25200.175.005.88715.07

223.25300.283.333.6518.8

345.25380.362.632.8728.43

467.25440.422.272.4447.12

589.25500.472.002.1647.58

61011.25550.521.811.9627.75

Sample calculationFrom the additional mass added (in kg), we need to first find the force (weight) produced by each additional load which means we need to,Weight = mass x gravitational acceleration

ExampleWeight = 2 kg x 9.81 m.s-2 = 19.62 NNext, for the time taken for 5 cycles of the system, we attained from the curve plotted by the mechanical recorder is in x (mm), to find the natural frequency of the system in both rad/s (wn) and cycles/s (fn), we use the formula

The theoretical value of spring constant, k, is equal to 1710 N/mThus, to get period, T

Example for experimental valueFrom the additional mass added (in kg), we need to first find the force (weight) produced by each additional load which means we need to,Weight=mass x gravitational acceleration ExampleWeight=2 kg x 9.81 m.s-2 = 19.62 N

Next, for the time taken for 5 cycles of the system, we attained from the curve plotted by the mechanical recorder is in x (mm), since the mechanical recorder rotates the graph paper by 20 mm/s, then to obtain the time taken for 5 complete cycles, t (s),

Thus, to get period, T,

Where, we then can calculate for frequency of spring-mass system, f,

Example

= 1.0 s

= 0.2 s

= 5.00 HzExample for theoretical value

*Where k = 1710 N/m

Percentage Error

Natural Frequency, fThe percentage error, % = | ftheoretical fexperimental | 100% ftheoretical = | 5.887 5.000 | 100% 5.887 = 15.0671 %

Spring constant, kThe percentage error, % =

= = 4.39%

8.0 ConclusionIn conclusion, we found that the experimental values of spring constant, k (N/m) is slightly different with the theoretical values. From the graph of Force (N) against extension of the spring (mm), we found that the values of slope is equal to the values of the spring constant by using the formula F=ks. The theoretical values of the spring constant is 1710 N/m while the experimental values of the spring constant is 1635 N/m. The values of the experimental and the theoretical slightly different due to the error that exist during conducted this experiment. The percentage error, % between the values of the experimental and theoretical is 4.39%.

In this experiment, the weight of frame is 1.25 kg. From the experiment, the values of the experimental frequency (Hz) for mass 1.25, 3.25, 5.25, 7.25, 9.25, and 11.25 is 5.00, 3.33, 2.63, 2.27, 2.00, and 1.81 respectively. For the theoretical values of normal frequency, f for mass 1.25, 3.25, 5.25, 7.25, 9.25, and 11.25 is 5.887, 3.561, 2.872, 2.444, 2.164, and 1.962. The values of the experimental and the theoretical a little bit different due to the error that exist during conducted this experiment. The percentage error, % between the values of the experimental and theoretical is 15.07, 8.8, 8.43, 7.12, 7.58 and 7.75 respectively. To improve the experiment result, Student should carefully put the additional weight during the experiment procedure as it would disturb the graph plotted. Besides, the pen/pencil should be put in properly tight to the hole in front of graph paper so that it will not harm the paper or if it too loose, it will slightly moving and disturbing the graph plotting. Moreover, the age of the spring very influence in the experiment results. The spring may not as elastic as stated by theoretical. Where, theoretical may represent the early state of spring where it is softer and more elastic. Otherwise, the spring is not actually homogenous during the deformation. After all, k theoretical is calculated based on the parameters where the spring is assumed to be homogenous during deformation as well the cross-section is uniform throughout the length of spring.