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# Determination of Force Constant k from Spring-mass System

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Introduction

Title: Determination of Force Constant k from Spring-mass System Objective: To measure the force constant k from a vertical spring-mass system Theory: A horizontal or vertical mass-spring system can perform simple harmonic motion. Let period be T, mass be m and force constant be k. Consider a vertical spring-mass system. Take downward as positive. Displacement of mass: x Static extension: e When it is in equilibrium: By Hook's law, With an arbitrary downward displacement x, the spring is stretched and there is a spring force pointing upward. The mass then accelerate upward. When the mass passes through the equilibrium position, there is no net force but with inertia, the mass goes upward and the spring is compressed then induce a downward spring force. The mass then accelerate downward and passes through the equilibrium position again. This process repeats itself if there is no other external forces. We can see that the displacement from equilibrium position always oppose its acceleration which means this is simple harmonic motion(SHM). ...read more.

Middle

7. Start the stopwatch and take the time t for 20 oscillations. 8. To reduce the random error, for each time adding one more slotted mass, take 3 sets of data for the period. 9. Calculate the time for one oscillation from the mean value of time for 20 oscillations. Plot a graph of T2 against m by Microcal Origin 6.0. Fit the points by the best straight line passing through the origin as if there is no mass hanged by the spring, there will not be any oscillation. 10. From the software, the slope M of the best fit line is calculated with error. The force constant k can be calculated by the equation: Results: The values of mass, time t for 20 oscillations and period square T2 (one oscillation) are recorded below: Mass/kg Time for 20 oscillations t/s Period square T2/s2 0.030100.00005 12.10.4 0.360.02 12.10.4 12.00.4 0.03990 13.80.4 0.470.03 13.70.4 13.80.4 0.049700.00005 15.40.4 0.590.03 15.30.4 15.30.4 0.059500.00005 16.80.4 0.710.03 16.80.4 16.80.4 0.069400.00005 18.20.4 0.820.04 18.10.4 18.10.4 0.078300.00005 19.20.4 0.920.04 19.10.4 19.10.4 ...read more.

Conclusion

So overall, the system does positive work on the surrounding and energy is taken away from the oscillations. This makes the period greater and so the force constant smaller. If the amplitude is too large, the damping force bv is than greater than the restoring force. This causes a much longer period. 3. As the slotted masses can be separated, during the oscillations, all slotted masses have inertia towards the two side end. When the object (masses and hanger) arrive to the highest position, it is just like projecting an object upward and it is free falling. This causes a larger force pressing downward and so a smaller T deduced. 4. Even though we use the G-clamp to fix the stand on the table, the table itself is still movable; the stick of the stand is movable also. So using G-clamp cannot assure that the system is not oscillating about one point. 5. Overall, the locus of the masses is not a vertical line but a circle, so the actual storing force in spring is only partly cancelled out by weight. ?? ?? ?? ?? ...read more.

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