**PURPOSE**

The purpose of this lab is to quantitatively investigate how the mass of a coil spring and the length of a pendulum affect the period of oscillation of a system experiencing simple harmonic motion. In this lab, I will establish a mathematical connection between the period and these variables through the execution of this experiment and data analysis.

**THEORY**

Simple harmonic motion or SHM is a type of periodic motion that can be found in a number of physical systems. These systems include springs and pendulums. To analyze and predict the behaviour of such systems, it is vital to understand the relationship between the oscillation period and the system’s parameters.

The key elements of simple harmonic motion that are vital to know for the completion of this lab are as follows:

- Equilibrium Position:
- The equilibrium position in SHM is the position at which the system is at rest, experiencing zero net force. For a pendulum, it is the vertical position when the string is vertical and the bob is at its lowest point. For a coil spring, it is the position where the spring is neither compressed nor extended.

- Period (T):
- The period of a system undergoing SHM is the time it takes to complete one full cycle of oscillation. It is the time interval required for the system to return to its initial position and velocity. The period is denoted by T and is measured in seconds (s).

- Length (l):
- In a pendulum, the length refers to the distance between the point of suspension and the center of mass of the bob. It plays a crucial role in determining the period of the pendulum’s oscillation. This lab will investigate the relationship between the length of a pendulum and its period of oscillation.

- Number of Oscillations:
- The number of oscillations refers to the count of complete cycles of motion made by the system. In the lab, we will measure the time for a specific number of oscillations to calculate the period accurately. Counting the number of oscillations allows us to determine the time interval required for multiple cycles.

**Pendulum:**

A pendulum is a physical system composed of a mass (bob) attached to a string or rod, which is fixed at one end. When the pendulum is displaced from its equilibrium position and released, it undergoes simple harmonic motion. The period of a pendulum is influenced by its length (l) and the acceleration due to gravity (g). According to the theory, the period (T) of a simple pendulum is given by:

T = 2π √(l / g)

This equation reveals that the period of a pendulum is directly proportional to the square root of its length. This experiment will investigate whether this is true or not.

Coil Spring:

A coil spring is a helical spring that stores potential energy when compressed or extended and releases it when allowed to oscillate. The period of a coil spring is influenced by its mass (m) and the spring constant (k), which represents its stiffness. According to the theory, the period (T) of a mass-spring system is given by:

T = 2π √(m / k)

This equation demonstrates that the period of a coil spring is directly proportional to the square root of the mass. This experiment will investigate whether this is true or not.

Period (T):

T = Total Time/Number of Oscillations

Percentage Error:

% error = (|measured – expected|) / (expected) x 100

Understanding these key elements is essential for conducting the lab experiment and analyzing

the relationship between the period of simple harmonic motion and the variables being investigated (pendulum length and coil spring mass). By manipulating these variables and measuring the resulting periods, we can observe how changes in length and mass affect the oscillatory behaviour of the systems.

**APPARATUS**

Materials:

- Pendulum:
- String

- Mass (bob)

- Stopwatch or timer

- Ruler or measuring tape

- Coil Spring:
- Coil spring

- Masses (with hangers)

- Stopwatch or timer

- Ruler or measuring tape

- Computer – PhetPhysics

**PROCEDURE: Part A: Dependence of Period on Pendulum Length (l)**

- Set up the pendulum:
- Attach a string to a fixed point and make sure it is vertical.
- Attach a mass (bob) to the other end of the string.
- Measure and record the length:
- Use a ruler or measuring tape to measure the length (l) of the pendulum from the fixed point to the center of the bob.
- Record the length in meters (m).
- Displace and release the pendulum:
- Displace the pendulum slightly from its equilibrium position and release it.
- Start the stopwatch as soon as the pendulum is released.
- Measure the time for a specific number of oscillations:
- Observe the pendulum’s motion and count the number of complete oscillations (back and forth).
- Stop the stopwatch when the desired number of oscillations is completed.
- Calculate the period:
- Divide the total time measured by the number of oscillations to obtain the period (T) of one oscillation.
- Record the period in seconds (s).
- Repeat steps 2-5:
- Repeat steps 2 to 5 for different lengths of the pendulum, ensuring a range of values.
- Graphical analysis:
- Plot a graph with the period (T) on the y-axis and the square root of the length (l) on the x-axis.
- Analyze the relationship between period and length using the graph.

**PROCEDURE: Part B: Dependence of Period on Coil Spring Mass (m)**

- Set up the coil spring:
- Hang the coil spring vertically from a fixed point, ensuring it is not stretched or compressed.
- Measure and record the mass:
- Measure and record the mass (m) of the coil spring.
- Record the mass in kilograms (kg).
- Attach masses to the spring:
- Attach a known mass to the coil spring using a hanger.
- Ensure that the mass is within the elastic limit of the spring.
- Displace and release the spring:
- Gently displace the spring from its equilibrium position and release it.
- Start the stopwatch as soon as the spring is released.
- Measure the time for a specific number of oscillations:
- Observe the spring’s motion and count the number of complete oscillations (up and down).
- Stop the stopwatch when the desired number of oscillations is completed.
- Calculate the period:
- Divide the total time measured by the number of oscillations to obtain the period (T) of one oscillation.
- Record the period in seconds (s).
- Repeat steps 2-6:
- Repeat steps 2 to 6 for different masses attached to the coil spring, ensuring a range of values.
- Graphical analysis:
- Plot a graph with the period (T) on the y-axis and the square root of the mass (m) on the x-axis.
- Analyze the relationship between period and mass using the graph.

**OBSERVATIONS**

Dependence of Period on Pendulum Length (l) | |||||

Length, l (m) | √Length (m) | Total time, t (s) | Number of oscillations | Calculations for Period | Period, T (s) |

0.5 m | 0.7071 m | 14.3 s | 10 | 14.3 / 10 | 1.43 s |

0.75 m | 0.8660 m | 18.1 s | 10 | 18.1 / 10 | 1.81 s |

1 m | 1 m | 20 s | 10 | 20 / 10 | 2.00 s |

1.25 m | 1.118 m | 22.4 s | 10 | 22.4 / 10 | 2.24 s |

1.5 m | 1.225 m | 26.3 s | 10 | 26.3 / 10 | 2.63 s |

Dependence of Period on Coil Spring Mass (m) | |||||

Mass, m (kg) | √Mass (kg) | Total time, t (s) | Number of oscillations | Calculations for Period | T (s) |

0.1 kg | 0.3162 kg | 10.8 s | 10 | 10.8 / 10 | 1.08 s |

0.2 kg | 0.4472 kg | 15.2 s | 10 | 15.2 / 10 | 1.52 s |

0.3 kg | 0.5477 kg | 19.4 s | 10 | 19.4 / 10 | 1.94 s |

0.4 kg | 0.6325 kg | 23.1 s | 10 | 23.1 / 10 | 2.31 s |

0.5 kg | 0.7071 kg | 26.7 s | 10 | 26.7 / 10 | 2.67 s |

**ANALYSIS**

Dependence of Period on Pendulum Length (l) Graph

In order to analyze the data this graph provides, a linear line of best fit must be added. This line will play a role in determining the correlation between the independent variable, in this case, √length (m), and the dependent variable, in this case, period (s). After determining the correlation value using online graphing resources, the relationship can be analyzed.

After finding the line of best fit for this graph, a correlation coefficient of 0.98904 was found. This value indicates a very strong positive correlation between the period and the square root of the pendulum’s length. The correlation coefficient measures the strength and direction of the linear relationship between two variables.

In the context of a “Dependence of Period on Pendulum Length” graph, a correlation coefficient close to 1 suggests that there is a nearly perfect positive linear relationship between the period (T) and the square root of the length (l) of the pendulum. As the length of the pendulum increases, the period also increases, and vice versa.

**Dependence of Period on Coil Spring Mass (m)**

In order to analyze the data this graph provides, another linear line of best fit must be added. This line will play a role in determining the correlation between the new independent variable, in this case, √mass (kg), and the dependent variable, in this case still period (s). After determining the correlation value using online graphing resources, the relationship can be analyzed.

After finding the line of best fit for this graph, a correlation coefficient of 0.99753 was found. This value shows an extremely strong positive correlation between the period and the square root of the mass of the coil spring. The correlation coefficient measures the strength and direction of the linear relationship between two variables.

In the context of a “Dependence of Period on Coil Spring Mass” graph, a correlation coefficient close to 1 suggests that there is an almost perfect positive linear relationship between the period (T) and the square root of the mass (m) of the coil spring. As the mass of the coil spring increases, the period also increases, and vice versa.

**Percentage Error Calculations**

Dependence of Period on Pendulum Length (l):

% error = (|measured – expected|) / (expected) x 100

% error = (|0.98904 – 1|) / (1) x 100

**% error = 1.096 %**

Dependence of Period on Coil Spring Mass (m):

% error = (|measured – expected|) / (expected) x 100

% error = (|0.99753 – 1|) / (1) x 100

**% error = 0.247 %**

**CONCLUSION**

In this lab, I verified the theoretical predictions by conducting experiments with pendulums of different lengths and coil springs with varying masses. I measured the mass/length, time, and the number of oscillations and used the formula **period = total time/number of oscillations** to measure the period of oscillation for each case. After finding the square root of both the lengths and the masses and plotting these values into two tables, I was able to gather data to analyze the relationship between the period and the respective parameters. I plotted the data on graphs and using the linear line of best fit and I determined the correlation value for both experiments.

For the period (s) vs √length (m) experiment, the correlation value between the two variables was 0.98904. This indicates that there is a very strong positive linear relationship between the period (T) of an oscillation and the square root of the length (l) of the pendulum. From this, it can be concluded that as the length of the pendulum increases, the period also increases, and vice versa. The percentage error for this experiment was 1.096%, which proves the theoretical predictions of the formula T = 2π √(l / g) that suggests that the period of an oscillation is directly proportional to the square root of the length of the pendulum.

For the period (s) vs √mass (kg) experiment, the correlation value between the two variables was 0.99753. This value indicates that there is a nearly perfect positive linear relationship between the period (T) of an oscillation and the square root of the mass (m) of the spring coil. From this, it can be concluded that as the mass of the spring coil increases, the period also increases, and vice versa. The percentage error for this experiment was 0.247%, which proves the theoretical predictions of the formula T = 2π √(m / k) that suggest that the period of oscillation is directly proportional to the square root of the mass of the spring coil.

The percentage errors for both experiments are so low because an online website was used to conduct the experiment. This online software allowed for the experiment to be conducted accurately with all of the length/mass and time values being precise. As a result, the correlation values for both the “Dependence of Period on Pendulum Length (l)” and “Dependence of Period on Coil Spring Mass (m)” display very strong linear relationships.

In conclusion, the objective of this lab experiment was to investigate the quantitative relationship between the period of simple harmonic motion, the length of a pendulum, and the mass of a coil spring. By recording and graphing the values of the period and the square root of the pendulum’s length, it was found that the period of a pendulum is directly proportional to the square root of its length. This confirmed the theoretical prediction. Similar results were found for coil springs, which showed that their period was directly related to the square root of their mass. Both the line of best fit and the high correlation coefficient values displayed that there were significant positive correlations between the period and the corresponding variables for the two experiments. These results emphasize the importance of length and mass in determining the period of oscillation in simple harmonic motion for both pendulums and spring coils. Overall, the findings support the predictions made by theory and offer insightful information about how these oscillatory systems behave.

Website used to conduct the **Dependence of Period on Pendulum Length (l)** experiment:

https://phet.colorado.edu/en/simulation/pendulum-lab

Website used to conduct the **Dependence of Period on Coil Spring Mass (m)** experiment:

https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.htm