Problem 67 - College Physics

Question

An astronaut on the distant planet Xenon uses an adjustable inclined plane to measure the acceleration of gravity. The plane is frictionless, and its angle of inclination can be varied. Here is a table of the data: $$\begin{array}{lc}\hline \theta & a\left(\mathrm{~m} / \mathrm{s}^{2}\right) \\\\\hline 5.0^{\circ} & 1.20 \\ 10^{\circ} & 2.49 \\\15^{\circ} & 3.59 \\\20^{\circ} & 4.90 \\\25^{\circ} & 5.95 \\\\\hline\end{array}$$ Make a plot of the measured acceleration as a function of the sine of the angle of incline. Using a linear "best fit" to the data, determine the value of \(g\) on the planet Xenon.

Step-by-Step Solution

Answer

The gravitational acceleration on Xenon, \(g\), is approximately 14.1 m/s².

Step 1: Calculate Sine of Angles

Given the angles of inclination \( \theta \), compute \( \sin(\theta) \) for each angle using the sine function.\[ \sin(5.0^\circ) = 0.0872, \\sin(10^\circ) = 0.1736, \\sin(15^\circ) = 0.2588, \\sin(20^\circ) = 0.3420, \\sin(25^\circ) = 0.4226 \]

Step 2: Prepare the Plot Data

Create a table combining \( \sin(\theta) \) values with their corresponding acceleration \( a \) values:\[\begin{array}{lc}\sin(\theta) & a(\mathrm{m/s^2}) \hline0.0872 & 1.20 \0.1736 & 2.49 \0.2588 & 3.59 \0.3420 & 4.90 \0.4226 & 5.95 \\end{array}\]

Step 3: Plot the Data

Create a scatter plot with \( \sin(\theta) \) on the x-axis and acceleration \( a \) on the y-axis. Plot each point from the prepared data table.

Step 4: Determine Line of Best Fit

Using a linear regression or a least squares method, fit a line to the plotted data points, which will have the form \( a = m \sin(\theta) + c \). In this context, \( m \approx g \) and \( c \approx 0 \).

Step 5: Calculate Slope as Gravitational Acceleration

Since \( a = g \sin(\theta) \) for an inclined plane with no friction, the slope \( m \) of the best-fit line corresponds to the gravitational acceleration \( g \). By calculating the slope from the best-fit line, we determine:\[ g \approx 14.1 \, m/s^2 \]

Key Concepts

Gravitational Acceleration

Gravitational acceleration is the rate at which an object speeds up as it falls freely towards a planet's surface. On Earth, this value is approximately 9.81 m/s². However, finding this value on other planets can be challenging due to varying factors such as mass and size of the planet. In our experiment on planet Xenon, gravitational acceleration was measured using an inclined plane.
This inclined plane method is effective because it slows down the acceleration, making it easier to measure. When the object travels down a frictionless incline, its acceleration depends on the angle of the incline and gravity. The experimental setup allows us to measure the acceleration of the object for each angle and derive gravity from these measurements.

• The formula used here is: \( a = g \sin(\theta) \)
• Where \( a \) is the acceleration, \( g \) is gravitational acceleration, and \( \theta \) is the angle of incline.

By rearranging and using measured values, we're able to calculate \( g \), the gravitational acceleration on Xenon.

Linear Regression

Linear regression is a statistical method used to find relationships between variables. In simple terms, it helps in finding the best straight-line equation to describe the data points on a graph. In the inclined plane experiment, linear regression is used to find the relationship between the sine of the angle \( \sin(\theta) \) and the measured acceleration \( a \).
The line of best fit calculated in this experiment shows how well the experimental data align with the mathematical model we assume. The linear equation has the form:

• \( a = m \sin(\theta) + c \)
• \( m \) represents the slope, which is equivalent to gravitational acceleration \( g \) in this context, and \( c \) is the y-intercept, which theoretically should be zero because no acceleration should occur without an incline.

By applying linear regression, we extracted the best estimate for \( g \), crucial for understanding the physics of planet Xenon.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle and are fundamental in understanding relationships in triangles. They are particularly helpful in physics experiments dealing with angles and inclines, such as with the inclined plane.
In this experiment, the sine function, \( \sin(\theta) \), indicates the vertical component of a sloped surface. Calculating \( \sin(\theta) \) allows us to understand how the incline affects the gravitational force on the object.

• The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.
• Values of \( \sin(\theta) \) are easily computed using a calculator for each measured angle to proceed with data analysis.

Understanding trigonometric functions grants insight into how angles influence motion and force, a crucial aspect of our inclined plane experiment.

Data Plotting

Data plotting is a key step in visualizing the relationships between variables in experiments. When data is plotted on a graph, it becomes easier to identify trends and calculate mathematical relationships like linear regression.
For the inclined plane experiment, plotting \( \sin(\theta) \) on the x-axis and acceleration \( a \) on the y-axis allows visualization of how changes in the angle affect acceleration. These data points create a scatter plot.

• Each plotted point represents a pair of measured values.
• A line of best fit can be added to illustrate the relationship more clearly and estimate further parameters, such as \( g \).

Being adept with plotting data helps in effectively communicating experimental findings and supports deeper analysis and understanding of the phenomena being studied.