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You are a scientist studying small aerosol particles that are contained in a vacuum chamber. The particles carry a net charge, and you use a uniform electric field to exert a constant force of \(8.00 \times 10^{-14} \mathrm{~N}\) on one of them. That particle moves in the direction of the exerted force. Your instruments measure the acceleration of the particle as a function of its speed \(v .\) The table gives the results of your measurements for this particular particle. $$ \begin{array}{l|cccccc} \boldsymbol{v} / \boldsymbol{c} & 0.60 & 0.65 & 0.70 & 0.75 & 0.80 & 0.85 \\ \hline \boldsymbol{a}\left(\mathbf{1 0}^{\mathbf{3}} \mathbf{m} / \mathbf{s}^{\mathbf{2}}\right) & 20.3 & 17.9 & 14.8 & 11.2 & 8.5 & 5.9 \end{array} $$ (a) Graph your data so that the data points are well fit by a straight line. Use the slope of this line to calculate the mass \(m\) of the particle. (b) What magnitude of acceleration does the exerted force produce if the speed of the particle is \(100 \mathrm{~m} / \mathrm{s} ?\)

Step by step solution

01

Plotting Data and Calculating Slope

To solve the first part, we graph the data using the given values for speed (on the x-axis) and acceleration (on the y-axis). We then fit a straight line to the data points. The slope of this line gives the relationship between acceleration and speed, and by the formula \(F=ma\) (where F is force, m is mass and a is acceleration), we aim to find the mass. The slope of the plotted line is mass, m, which is calculated by taking the force \( F = 8.00 \times 10^{-14} N \) divided by the slope.

02

Conjecture for Mass of the Particle

Once we calculate the slope and have the value of force, the mass of the particle can be calculated as \( m = F / slope \). It is important to note that the units of the calculated mass should be in kilograms to maintain consistency with SI units.

03

Computing the Magnitude of Acceleration

For part (b), given that the speed of the particle is \( 100 m/s \), and already knowing the mass of the particle from the previous step, we can now compute the magnitude of acceleration that the exerted force creates. This is done again using the formula \( a = F/m \) , by substituting the known values for force and mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aerosol Particles

Aerosol particles are tiny solid or liquid fragments that are suspended in a gas, typically air.
When studying them in a vacuum chamber, we remove air to eliminate variables such as drag or friction. This allows us to concentrate on the charged particles’ behavior when influenced by an electric force.
In this experiment, aerosol particles carry a net electric charge, making them reactive to electric fields. This behavior highlights the principle of electric force: charged objects within an electric field experience a force proportional to their charge and the electric field's strength.
Understanding how aerosol particles move and respond under controlled electric fields helps in fields like air purification and climate studies, where particle behavior is crucial.

Acceleration

Acceleration refers to the rate at which an object’s velocity changes over time. It can be the result of a force acting upon the object. In our study, the electric force causes the aerosol particles to accelerate.
The relationship between force and acceleration is governed by Newton's second law of motion: \( F = ma \). This means that for a given force, the acceleration experienced by an object is inversely proportional to its mass.
In the given exercise, the acceleration of the aerosol particle decreases as its velocity increases. This suggests a negative correlation, indicating that the particle might be reaching limits imposed by other factors, like relativistic effects, which impact particles traveling at significant fractions of the speed of light.

Particle Mass

The mass of an aerosol particle plays a crucial role in determining how it responds to forces. In this study, measuring the mass is essential since it affects acceleration during motion.
To find the mass, we utilize the slope of the line fitted to the plotted data of acceleration against speed. The slope reflects the rate of change of acceleration with respect to speed, linked directly to mass by rearranging the equation \( F = ma \) into \( m = F/slope \).
Given the force is constant, a steeper slope indicates a larger mass while a shallower slope suggests smaller mass. Calculating this accurately is vital for experiments that depend on precise mass measurements, such as those in material science and aerosol behavior analysis.