How Does Graphing Time Squared Reveal Pendulum Dynamics?

[QuarkCharmer]

Homework Statement

We did a lab, where, in the first part we timed the period of a pendulum with varying lengths. In the second part, we timed the period of a pendulum with varying masses.

I got the results that I expected to get. However, I do not understand two of the lab questions.

A.) Length: Go to Graphical Analysis and Graph Time vs. Length. Explain the effects of length on the time of the swing.

B.) Go to Graphical Analysis and Graph Time vs. Mass. Explain the effects of mass on the time of the swing.

Homework Equations

$$T_{p}=2π\sqrt{\frac{L}{g}}$$

The Attempt at a Solution

Clearly, mass had no effect on the time of the swing, and as the length of the string increased, so did the period of oscillation.

What I don't understand is what I am supposed to graph exactly? For both of those parts, the professor instructed us to use a graph of time squared vs. length and time squared vs. mass. and then gave us a subtle hint to solve for g.

I don't understand what she is getting at with this cryptic clue. What is the significance of graphing the data with time squared? How am I supposed to find our local gravitational constant from this graph? (Assuming that is what was implied).

Thanks

 

[PeterO]

Homework Statement

We did a lab, where, in the first part we timed the period of a pendulum with varying lengths. In the second part, we timed the period of a pendulum with varying masses.

I got the results that I expected to get. However, I do not understand two of the lab questions.

A.) Length: Go to Graphical Analysis and Graph Time vs. Length. Explain the effects of length on the time of the swing.

B.) Go to Graphical Analysis and Graph Time vs. Mass. Explain the effects of mass on the time of the swing.

Homework Equations

$$T_{p}=2π\sqrt{\frac{L}{g}}$$

The Attempt at a Solution

Clearly, mass had no effect on the time of the swing, and as the length of the string increased, so did the period of oscillation.

What I don't understand is what I am supposed to graph exactly? For both of those parts, the professor instructed us to use a graph of time squared vs. length and time squared vs. mass. and then gave us a subtle hint to solve for g.

I don't understand what she is getting at with this cryptic clue. What is the significance of graphing the data with time squared? How am I supposed to find our local gravitational constant from this graph? (Assuming that is what was implied).

Thanks

Graphing time squared is essential.

presumably when you graph time vs length you get a curve - but what curve?

If it curves up, it could be y= x²; y = x³; y = tanx

If it curves the other way, it could be the start of y= sinx ; y = √x ; y = ³√x

The only graph you can interpret with confidence is a straight line.

graphing the square of time against length might yield a straight line.

 

[QuarkCharmer]

I understand that it straightens out the curve into a line, so you can then find a best fit, or calculate the slope of that line, or just read the data. What I don't understand is how this length v. time^2 graph can be used to find the local gravitational constant.

I found the slope of the graph to be about 4 point something or other, and I have been trying to figure out how I can use that for anything relating to the goal of this experiment.

 

[I like Serena]

Hi QuarkCharmer!

From your relevant equation it follows that:
$$T_{p}^2=\frac {4\pi^2} {g} L$$
This means you should find a straight line through the origin.
The slope is $\frac {4\pi^2} {g}$
So:
$$g = {{4\pi^2} \over \text{4 point something or other}} = 9.81$$

See how nicely it fits the actual acceleration of gravity!

 

[asteriatic]

for your question. It seems like the lab was investigating the relationship between the period of a pendulum and its length and mass. In order to fully understand the effects of these variables, it is important to plot the data in a way that highlights the relationships.

For part A, you are asked to graph time vs. length. However, instead of just plotting time and length, you are asked to plot time squared vs. length. This is because the equation for the period of a pendulum includes the term for length squared (T=2π√(L/g)). So by graphing time squared vs. length, you will be able to see a linear relationship between these two variables. This will make it easier to analyze and understand the effects of length on the period of the pendulum.

Similarly, for part B, you are asked to graph time squared vs. mass. This is because the equation for the period of a pendulum also includes the term for mass (T=2π√(L/g)). By graphing time squared vs. mass, you will be able to see the relationship between these two variables and how they affect the period of the pendulum.

To find the local gravitational constant (g), you can use the slope of the linear relationship in both graphs. The slope represents the square root of g. By squaring the slope, you can find the value of g. This is because the equation for the period of a pendulum can be rearranged to solve for g as g=(4π^2/l) * (slope)^2.

I hope this helps clarify the purpose of graphing time squared vs. length and mass in this lab. By doing so, you will be able to better understand the relationships between these variables and how they affect the period of a pendulum.