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Copland CollegeYear 11 - Physics |
Equations of Motion: Reading: Chapter 3 - Physics - Principles and Problems: Our focus up until this point has been the analysis of graphs showing objects moving in some uniform way. This uniformity could involve position, velocity or acceleration. The motion described in the graphs can also be described by using equations. There are four basic equations which we will use. These are shown below.
It is important to note that the symbols used to describe the various quantities varies according to the text that you are using. It is essential that you become familiar with all the permutations. A table with the variety of symbols that are used is shown below.
This list is not exhaustive and can include other possible variations. One of the major traps for those new to equations of motion is to use the equation v = d/t for motion that has a non zero uniform acceleration. Usually the problem asks something like the following. A racing car starts from rest and is uniformly accelerated to 41 m/s in 8 seconds. What is the car's displacement. The equation can be altered to make d the subject. Therefore d = vt Unfortunately this equation cannot be used because there is a uniform acceleration. The way to approach this problem is to calculate the average velocity that this car experiences and then multiply by the time taken. The average velocity is calculated by taking the initial velocity and adding to this the final velocity and then dividing this sum by 2. That is d = (vf + vi)/2 x t An excellent discussion on the derivation of the equations shown above can be found at the following site. http://www.physchem.co.za/Graphs/Equations.htm The Simple Pendulum: A simple pendulum consists of a mass hanging from the end of a string. The string is considered to be of negligible mass in comparison to the system as a whole. It was stated that the period (T) of the pendulum can be described by the formula,
The period is the time for one complete swing. That is the time it takes to return to your hand after it is released. l is the length of the system in metres and g is the acceleration due to the Earth's gravity. The interesting features of this equation, apart from including the term Pi (p ), is that we can use the pendulum to measure the acceleration due to gravity, g. The value of the acceleration due to gravity on Earth is about 9.81 m/s/s. The general average for Earth is taken as 9.80665 m/s2 This means that any object dropped on Earth will accelerate towards the Earth's surface in such a fashion that it increases its speed (velocity) by 9.81 m/s every second. This means that after 2 seconds it is travelling 2 x 9.81 m/s/s, i.e., 19.62 m/s/s and after 3 seconds it is travelling at 3 x 9.81 m/s/s, i.e., 24.93 m/s/s. Of course objects as they fall are always significantly affected by the air they fall through. Air resistance, as it is called, effectively means that it is very difficult to accurately determine the acceleration due to gravity by timing the fall of an object. Most object reach a terminal velocity very quickly. In class we used the equation above in a slightly different from. We measured the period of the swing, we knew the length of the pendulum, in metres and it was possible to determine the value of g. The equation has the form,
The practical that you are completing will be assessed primarily on data collection and data processing. Data Collection: The two variables associated with this activity are the length of the pendulum and the period of the swing. The length is the independent variable and the period is the dependent variable. Normally the independent variable is plotted on the horizontal and the dependent plotted on the vertical axis. This is a general convention which will be reversed in this case such that in doing so will allow the slope of the line to be a direct measure of the acceleration due to gravity. For any experiment you will need to have all other variables such as mass of pendulum bob and angle of swing under very close control. You need to keep in mind that the formula that you are using is an approximation. Also what role does uncertainty play in this activity? Does the uncertainty associated with the measurements really allow you to achieve an accurate value for the acceleration due to gravity? The first series of experiments will focus on varying the pendulum's length keeping mass and swing angle constant. At least 8 sets of data should be collected. 10 if time is available. From this series of experiments the value of the acceleration due to gravity can be calculated using graphical techniques. You should use Excel for these graphs. Use Excel to generate an equation for the line but make sure that you force the line to go through the origin, i.e., with zero length the period of swing must also be zero. The second series of experiments will involve keeping swing angle the same as series 1 but the mass will be varied. The length must be a value used in series 1 experiments. The period of the swing will be then determined. The graph to plot here is mass (horizontal axis) vs period of swing. The line should be horizontal. The third series of experiments will involve keeping the mass and length constant. Choose values that you have used in the previous two sets of experiments. The angle of swing will now be varied. Don't choose an angle that is too large. Maximum angle should be about 90 degrees from vertical. Plot graph of angle of swing (horizontal axis) vs period. Look very carefully at the line produced. The line produced may not be a straight line. Data Analysis Using Microsoft Excel: In the report that you produce you will focus on data collection and data presentation. The data presentation can be most successfully done using Excel. The graphs are then produced in Excel directly from this data. Series 1 Experiments - Relationship between length of pendulum and period of swing: The following diagram shows one possible way to present the data for Set 1 Experiments.
In the data above only a single set of data was collected for each length of pendulum. Some people did multiple experiments and this would require that the data for each experiment be averaged. If two experiments were done for each pendulum length then the spreadsheet might look like the following.
The formulas used to calculate the various columns are shown below.
The Period Average (T) column is calculated by averaging the data in columns B and C and then dividing by 10. The dividing by 2 and then dividing by 10 means a division by 20. Remember all cells other than your data cells need to be calculated via an Excel formula. Don't do calculations via a calculator and then just type in your answers. The first graph that you should draw will look something like the following,
To produce this graph from the data above or any data you will need to follow the the steps as indicated. Highlight the 4Pi^2l
and the Period^2 columns. Remember that in Excel the
symbolism Pi^2 means Pi Squared. Then click the
Click the Next box and step 2 of the wizard will appear.
You will note that the X and Y axes are the wrong way around. The reason that they should be the other way is that then when we calculate the slope it will give us a direct measure of the acceleration due to gravity. As the graph looks now we will get the inverse of the acceleration due to gravity. To reverse the axes proceed as follows. Click the Series tab.
Now click the red arrow in the X Values box. The following wll appear.
You will notice here that the 4Pi^2l column is highlighted. We want T^2 to be the X values. To select the T^2 column to be the X values drag the mouse across the T^2 column. Don't include the title cell. See diagram below.
Now click the red arrow and repeat the process with the Y Values: After you have completed this click the Next button and the following dialog box will appear.
Choose an appropriate title for your graph and label the axes. Click the Axes tab but leave the options unchanged. For the Gridlines check the Value (X) axis Major gridlines.
From the Legend tab uncheck Show Legend. Data Labels remains as is. Click the Next button. Choose a location for your chart.
It is probably best to choose As new sheet , at least initially. You can always go back later right click on the graphs and change this location. See below
The graph that is produced will look similar to the following,
except you won't as yet have added the trend line and the equation. The R^2 represents the correlation coefficient. The closer it is to ONE the better the data fits the line of best fit that we have generated. To add you trend line and equation right-click on one of the data points. The following dialog box will appear,
Choose Add Trendline. When the dialog box appears choose the Linear option
Now click the Options tab. Tick the options as shown below.
Highlight the equation and then right click on it. Change the font size and colour (if you wish) Right click on the trend line and format its colour to red. Series 2 Experiments - Relationship between mass and period of swing. Typical data for this set of experiments might look like the following.
The formulas used to generate this table appear below.
The graphs that should be produced is as follows.
This graph can be generated in the way discussed previously. Note the interesting correlation coefficient. Remember our pendulum formula shows how the mass of the pendulum should have no effect on its period. Series 3 Experiments - Relationship between swing angle and period of swing: Some typical data for this experiment is as follows,
The graph generated from this data is shown below.
The trendline here is not linear but rather polynomial. See the dialog box below.
Of course your data may fit some other type of curve. Investigate the curve fit until the best one is found. If the angle of swing had no effect then you would expect to find a horizontal line. The line produced with this particular set of data suggested that as the angle increased the deviation of the observed results from the theoretical results tended to increase. Uncertainty Analysis - Error Analysis: Determination of the uncertainty can entail some very detailed calculations. However in this case the approach that you should take is as follows. Each data point on the main graph will have a maximum value and a minimum value as determined by the experimental uncertainty.
These maximum and minimum values can be represented by drawing error bars on your graph. Although Excel is able to do this we will do in manually. See the end of this section for a method that uses Excel. The error bar is a vertical line drawn through each point on the graph. It represents the range of expected minimum and maximum value. We now need to determine the uncertainty associated with the quantities that we were required to measure, i.e., time and length. For the timing the uncertainty was minimised by timing over 10 swings. A value of 2 x 0.2 seconds (0.4 seconds) over 10 swings is probably reasonable. The two values reflect the error at the start and at the end. Thus over a single swing it will be plus or minus 0.04 seconds. With respect to measuring the length of the pendulum we will consider that the uncertainty is probably in the order of plus or minus 0.5 cm (5mm). Thus a length measured at 1.91 m could range between 1.905 and 1.915metres. You can now rework you calculations considering a maximum length combined with maximum period and minimum length combined with minimum period. Part of a spreadsheet showing this is shown below.
The complete spreadsheet shown above can be downloaded by clicking here. Using Excel to generate Error Bars: Excel can produce error bars on your graph. You are able to choose how these error bars are generated. The options are displayed in the following dialog box.
You are able to generate both X-error and Y-error bars. The error bars can be generated in a variety of ways. We would choose the Standard Error method. What this does is to calculate the standard deviation of the Y or X data and then divide it by the square root of the number of measurements. This division reflects the fact that we expect the uncertainty of the average value to get smaller when we use a larger number of measurements. As an example the following instructions will allow you to generate Y-error bars. We will use the following graph for this example.
Right click on one of the data points. All the data points in the series should be displayed. You will probably need to have your graph in full page view so that the points are easy to select. For this choose its Location as aAs new sheet. The following dialog box should appear.
Choose the options as shown in the Format Data Series dialog box shown previously. The error bars will appear as shown below.
If you wish you can also generate X-error bars. Further Investigations: To investigate some other aspects of a pendulums motion we can view some web-based simulations. Click on the following link, http://www.walter-fendt.de/ph11e/pendulum.htm Here you should see a Java based simulation. Accept the default values as shown.
Read the introduction carefully then carry out the following investigations. Make sure that you return the settings to the default values between each activity. To do this press the Refresh icon on the toolbar or press F5 Q4) Double then triple the mass. What happens to the oscillation period? Q5) Double then triple the length. What happens to the oscillation period? Q6) Double then triple the amplitude. What happens to the oscillation period? Q7) The gravitational field on the Moon is 1.62m/s/s and on Jupiter it is 23.54 m/s/s. What happens to the period of oscillation a pendulum of equal mass and equal length placed on the surface of these two bodies? Free Fall: In the following activity you will examine a simulation of what happens to objects as they fall under the influence of the Earth's gravitational field. As you may recall objects on Earth all fall with an acceleration of 9.81 m/s/s. This means that after 1 second the object has increased its speed to 9.81 m/s/s, after 2 seconds this speed has increased to 2 x 9.81 m/s/s = 19.62 m/s/s. Unfortunately the resistance caused by the air has a considerable effect on how objects really fall. This is what you will investigate today. Click on the following link or the picture, http://www.hazelwood.k12.mo.us/~grichert/explore/dswmedia/interact.htm You should see the following,
Should the Freefall Lab not appear click on the Mechanics link and then click the Freefall Lab. You will need to print the the following worksheet or obtain a copy during class time. Part 1. Increasing
the Mass of an Object without Air Problem #1: What effect does the mass of a falling object have on its time of fall? Independent Variable: Mass of
falling sphere - Variable under your control. The graph displays Acceleration in Green, Distance fallen in Blue, and Velocity in red. Using your mouse you can place the arrow into the graph grid-area and use the smart (+) cursor to determine the x and y coordinates of the data. Remember the y- coordinate is the vertical distance and the x - coordinate is the horizontal coordinate. Use the smart cursor to find the final speed, time of fall, and acceleration. Clear the trails before each trial. Complete the table below:
1)What is the effect of more mass on the time of fall in a vacuum? ___________________ 2) What was the effect of more mass on the acceleration in a vacuum? ________________ 3) What was the effect of more mass on the impact speed? __________________________ 4) Under these ideal conditions (in a vacuum) we would state that the time of fall is __________________ (dependent or independent) of the mass of the falling object. 5) Which graphed lines were straight lines? ______________________________________ 6) Which quantities were changing on the graph? _____________________________________ Part 2 - Increasing the Mass of an Object with Air? Set the Slide controls as follows: Anything not
specifically mentioned keep the same as before. Problem #2: What effect does the air have on the time of fall. The data below will be compared to the previous data to determine the effect air has on the time of fall for objects of different mass. Independent Variable: Mass of
falling sphere -
7) What effect did air have on falling time compared to a vacuum? ________________________ 8) Which mass had the shortest time of fall? __________________ Further Investigations: Problem 1: Investigate
the effect of dropping a ball from 100 metres with air resistance
off and when set to 0.2. Set the time scale to 8 seconds,
change the vertical scale to range from -10 to 100. Do not clear
trails between runs. Collect data to show what happens to the
velocity (speed) and acceleration during each fall. Write a
paragraph discussing the results. Remember how to interpret the
slope of these particular graphs of distance travelled, speed and
acceleration. Vectors: Text Reference: Chapter 6 - Physics Principles and Problems: We have already come across the concept of quantities such as displacement, velocity and acceleration requiring that they be defined with a magnitude as well as a direction. For example to completely define the velocities of objects travelling in the opposite direction it is necessary to define one direction as positive (+) and the opposite direction as negative (-). Force is also a vector quantity. To define any force completely it requires a magnitude, for example 3 Newtons, 10 Newtons etc., as well as a direction, for example 45 degrees East, (+) positive or South etc. The magnitude and direction of a particular vector quantity are generally represented by ARROWS attached to the object. The length of the arrow represents the magnitude of the vector quantity, for example an arrow of 5 cm in length might represents a 5 Newton force, and where the arrow points represents the direction of the applied force To demonstrate the process of adding vectors together graphically to produce a RESULTANT force click on the following diagram
At this site you will be able to add various force vectors together and observe how a RESULTANT force is obtained. The length of the vector arrow represents the magnitude of the force and where it is pointing represents how the force is acting on the object. Another site that demonstrates the addition of vector quantities using pictorial scale diagrams is shown below. Click on the diagram to visit this site.
The explanation of this simulation is as follows. The black line can be considered the main vector. THis vector is thyen resolved into a horizontal, red, component and a vertical, blue, component. Alternatively the red and blue vectors can be considered as being added together vectorially. This produces a resultant which is represented by the black line. A good discussion of vectors can be found by clicking the following, http://id.mind.net/~zona/mstm/physics/mechanics/vectors/introduction/introductionVectors.html Velocity and acceleration vectors are shown by visiting the simulation shown below Click on the diagram to take you to the site running this simulation. At this site you will notice that you can alter the initial position, velocity and acceleration of the object. You will also notice that graphs of these three quantities vs time are shown. You will also notice that you can display the vector quantity of either velocity or acceleration. This vector quantity is represented by an arrow that is attached to the car. The length of this arrow represents the magnitude of the velocity or acceleration and where the arrow points represents the direction of this velocity or acceleration.
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icon on the menu bar. This will bring up the Chart
Wizard dialog box. See diagram below.
