Harmonic Motion

Old Dominion University
PHYS 111N
Experiment 10 Harmonic Motion
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Introduction In this experiment we will investigate the simple harmonic motion of an object suspended by a spring that oscillates on a vertical plane and in a separate experiment was examine oscillations on a horizontal plane. In simple harmonic motion, the displacement from the equilibrium position is directly proportional to the force. The force generated is always directed toward the equilibrium position. If the object is at its vertical peak and descending, the force is directed downward toward the point of equilibrium. The same is true for the objects in a vertical system or a horizontal system. Because the force always is directed towards the equilibrium position it is referred to as the restoring force. The spring constant of the springs used in the experiment must be calculated prior to performing any additional step. This is done using the Pasco Scientific Data Studio and equipment. We set up the 36” vertical support rod on the table clamp and attached the 90o adapter. The 24” support rod was set and the force sensor was place on the 24” rod. Once the force sensor was calibrated, a sequence of masses was suspended from the spring and the amount of stretch was measured and documented in Table 1 as spring 1. The graphing program calculated the slope of the results equaling to the spring constant, Graph 1. This process was duplicated for the second spring (spring 2) and documented in Table 1. Part A investigated the vertical oscillation of a 0.75kg mass suspended from the spring. The mass was lowered approximately 5cm towards the motion sensor and then gently released. The DataStudio program generated the change in position versus time and Graph 2 displays the results. The results gave a harmonic wave and seven consecutive high points were documented in Table 2. Part B investigated horizontal oscillation using two springs attached to the collision cart, one on each end of the cart. The fence was placed on the cart and their combined mass was documented in Table 3. The fence was mounted on the cart and photogate was mounted so the beam was broken by the single opaque bar. The cart was then drawn approximately 20cm to the right from its equilibrium and was released to allow the springs to create a horizontal oscillation. The DataStudio program recorded the time period of the oscillations where the mean was the measured period, the slope=Tmeas. The same process was carried out with the compact mass bar on the cart. Both results were documented in Table 3.

Data

Table 1
Spring Constant Totals Mass on Spring 1 (m) | 0g | 20g | 30g | 40g | 50g | 60g | 70g | Spring Length (x) | 0.092m | 0.015m | 0.044m | 0.078m | 0.11m | 0.143m | 0.174m | Spring Constant (k1) | 1.09 N*m | | Total Mass on Spring 2 (m) | 0g | 20g | 30g | 40g | 50g | 60g | 70g | Spring Length (x) | 0.092m | 0.043m | 0.071m | 0.102m | 0.130m | 0.156m | 0.178m | Spring Constant (k2) | 11.4 N*m | |

Table 2
Vertical Oscillation High Point # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Position | 0.404m | 0.403m | 0.402m | 0.398m | 0.398m | 0.402m | 0.402m | Time | 1.0956s | 2.1414s | 3.1373s | 4.1831m | 5.1791m | 6.1751m | 7.1711m | Period | | 1.046s | 1.0233s | 1.0458s | 0.996s | 0.996s | 0.996s | Total Mass on Spring (m) | 0.075 | Spring Constant (k1) | 1.09 N*m | Average Measured Period of Oscillation (Tavg) | 1.017s | Theoretical Period of Oscillation (T) | 1.65s | % Difference (Theoretical Period Vs. Average Measured Period) | 38.4% |

Table 3
Horizontal Oscillation Item | Value | Item | Value | Mass of Cart (With Picket Fence) | 0.51479kg | Spring Constant, k1 | 1.09 N*m | Mass of Compact Cart Mass Bar | 0.4952kg | Spring Constant, k2 | 11.4 N*m | Run #1 (Cart without Compact Mass Bar) | Run #2 (Cart and Compact Cart Mass Bar) | Measure Period (slope) (Tmeas) | 1.7767s | Measure Period (slope) (Tmeas) | 2.4622s | Theoretical Period (T) | 1.26s | Theoretical Period (T) | 1.787 | % Difference | 41.0% | % Difference | 37.8% |

Data Analysis The spring constant of a spring can be calculated utilizing Hooke’s law where the spring constant is directly proportionally to force and inversely proportional to the distance stretched. . In Part A we were requested to calculate the spring constant given a known mass and measured stretch of the spring. Using the prior equation we calculated the spring constant using the average of the calculations for the spring constant of the seven trials. For the vertical oscillation experiment the period of oscillation depends on the spring constant that was calculated in Part A. Using the formula . We are able to measure the oscillation using the mass, spring, and force sensor. We can then verify the relationship between the calculated spring constant by comparing the period of oscillation. We were able to get within 14% of the measure value. All calculations are present on attached forms. The final portion of the experiment measured horizontal oscillation. This has to take into account the affects of two springs attached to the cart that was used on the track. To take into account the additional spring in the system we needed to alter the equation used for calculating period. The new equation was used. The two springs act as closed system. The only forces acting on the cart in an ideal environment are gravity and the force generated by the springs. As one spring’s potential energy is released and converted into kinetic energy, the other spring is acting in concert but inversely in direction and magnitude. We calculated the difference of the calculated and measured to be under 10 percent in both springs.
Conclusion
Overall the experiment was a great success in demonstrating the mathematical and physical relationship between a spring constant and the mass on which it acts. The period of oscillation was determined to be within 10% of measured and calculated values. The difference may be due to friction of the air and cart resulting in a loss of momentum. This loss was minimal, but present none the less.
Analysis Questions 1. When the position of the mass is farthest from the equilibrium position, what is the relative velocity of the mass? a. Because the masses are in static equilibrium there is no acceleration. 2. When the absolute value of the velocity of the mass is greatest, where is the mass relative to the equilibrium position? b. When it is at the mid-point between the equilibrium position and the lowest point 3. At what point(s) was/were the system’s energy entirely kinetic? At what point(s) was/were the system’s energy entirely potential? c. When looking at vertical oscillation, the energy was entirely kinetic when the mass attached to the spring was at its highest velocity and was entirely potential when the spring was completely stretched. The same is true for horizontal oscillation. However, in horizontal oscillation the potential and kinetic energy were split between the two springs. As the cart moved on the track the ratio of kinetic and potential energy was exchanged between the springs. 4. A spring is attached to an overhead beam. If you put a mass of 170g on it, and it proceeds to stretch a distance of 7cm, what is the spring constant? Using the same spring, you apply a force that stretches the spring a distance of 20cm. What is the magnitude of the downward acceleration? (Use units of m/kg/s) Show your work. d. SEE ATTACHED 5. A block is connected between two springs on a frictionless, horizontally level track. The other end of each spring is respectively anchored to a vertical pin set at opposite ends of the track. If the spring constants are determined to be 3.00N*m and 1.6N*m, and the period of oscillation for the block when slightly displaced from equilibrium point is 0.600s, what is the mass of the block? Show your work. e. SEE ATTACHED