To prove that the period of a mass-spring oscillation system is given by: 2π(m/k)1/2.
THEORY:
Using Newton's second law, we were able to formulate a differential equation in the form of a = -(k/m)1/2x. In this form, the constant in front of the x is equal to ω2, the angular frequency of the system. Once we found ω, we plugged it into the original equation for the period: T = 2π/ω, or: 2π(m/k)1/2.
APPARATUS:
Our apparatus consists of a ring stand, C-clamp, a metal rod, a spring, and a hanging mass.
EXPERIMENTAL PROCEDURE:
Each group was given a different spring with an unknown spring constant, k. We used the setup pictured above to find the displacement of the red indicator on the spring hook. This allowed us to use Hooke's Law to find the spring constant. Furthermore, we also used a smartphone timer to measure the period of the mass-spring system using various values for m: the effective oscillating mass of the system (given by Mhanging + Mhook + Meff of spring = m→115 g initially).
 |
| Spring Constant Calculation |
 |
| (1) Period Data for our group's spring |
 |
| (2) Period Data for every group's springs |
 |
| Spring Constant, k vs period for every group (constant effective oscillating mass) |
 |
| Our group's period data (differing effective oscillating masses) |
 |
| Effective oscillating mass vs period (our group's system) |
ANALYSIS:
After finding the spring constant of our spring using Hooke's Law, we measured the period of our system using a stopwatch. This allowed us to create curve fits of the period vs effective oscillating mass and period vs. spring constant graphs. We decided that a curve fit would be the best comparison to our derived equation for the period, 2π(m/k)1/2, because Logger Pro would give us an equation in the form of A*z^B, where A is a constant containing independent variables and other constants, z is our dependent variable, and B is the power to which the dependent variable was raised. For example, in the curve fit of the period vs. spring constant graph, A would equal 2π(m)1/2, because it only contains the independent variable, m, and other constants, while B would equal -1/2 because it is the power to which the dependent variable, k, is raised to in our theoretical equation for the period of the system. Additionally, our data demonstrates that the period decreases as the spring constant k increases. This is consistent with our theory because as k increases, the stiffer the spring should be. Physically, this means that the spring applies a greater magnitude of force per meter in order to counteract the force of gravity, thereby decelerating the system faster than with a lower k. Moreover, our data shows that the period should increase if you increase the mass attached to the spring. Once again, this is consistent with our theory because a greater mass means a greater force of gravity. Assuming our k is constant, this signifies that the system should take a longer time to decelerate because the force of gravity has increased but the rate at which the spring counteracts the force of gravity has not.
CONCLUSION:
Our lab results were not as successful as we had hoped. On the one hand, we were able to accurately predict the relationship between the period and the spring constant. For example, in our first curve fit, we found A = 2.022, which was very close to our predicted value of 2.13---5% difference---and we found B = -0.4727, which was also very close to our predicted value of -0.5---also a 5% difference. On the other hand, we were extremely inaccurate at predicting the relationship between period and the effective oscillating mass. For instance, the curve fit of the aforementioned graph reveals that A = 0.01694 when our prediction found A = 1.646 --- 99% difference --- and B = 0.7323 when it should have been 0.5 --- a 46% difference. This uncertainty in our calculation was most likely due to our values of m being relatively small. At the minute increments our group increased the mass by, it is possible that the trend we found in our data is not truly representative of the scope of this experiment.
No comments:
Post a Comment