Newton's second law, F = ma, predicts the future
motion of an object if we know its initial motion and the net force that is
acting on the object. Today you will explore this
motion when the net force is approximately constant.
Equipment needed:
Tennis ball (or other ball)
Open a Microsoft Word document to keep a log of your
experimental procedures and your results. This log will form the basis of
your studio session report. Address the points highlighted in blue.
Answer all questions.
Free-fall - The acceleration of gravity
Gravity is the force of nature we are most aware of.
One can argue that other forces, such as
the electromagnetic force, which holds molecules together in solid objects, or nuclear forces, which determine the structure of atoms, are more important,
but these forces are less obvious
to us. Near the surface of Earth the force of
gravity on an object of mass m equals F_{g} = mg. It is
constant and points straight down. If we can neglect other forces and the
net force is approximately equal to F_{g}, then we have motion
with constant acceleration g.
Observation
Hold a tennis ball at about your height and then let go. Observe the
motion of the ball.
Describe the motion as the ball is falling.
Estimate how long it takes the ball to reach
the floor.
What can you say about the speed of the ball
as a function of the distance it has already fallen?
If you drop the ball from about half of your
height, does it take approximately half the time to reach the floor?
Experiment 1
It does not take the ball a long time to reach the floor.
It is hard to get detailed information about its motion without using external
measuring instruments. In this experiment, the instrument is a video
camera. You will analyze a video clip. The clip shows a ball being
dropped. You will determine the position of the freely-falling
ball as a function of time by stepping through the video clip frame-by-frame and
by reading the time and the position coordinates of the ball off each frame.
You will construct a spreadsheet with columns for time and position and use this
spreadsheet to find the velocity as a function of time. The slope of a
velocity versus time graph yields the acceleration of the ball.
Procedure:
To play the video clip or to step through it frame-by-frame click the "Begin"
button.
"Play" the video clip. When finished, "Step up" to
frame 1. In some browsers you have to click "Pause" first.
In the setup window, choose to track the y-coordinate of an
object.
Click "Calibrate". Then click "Calibrate Y". The video clip contains
a meter stick. Position the cursor over bottom end of the stick and click the left mouse button. Then position the cursor
over the top end of the stick and click the left
mouse button again. This will record the y-coordinates of the chosen
positions. Enter the distance between those positions into the text box in
units of meter. For the example positions, you would enter 1 into the
text box. Click "Done".
Make sure the video frame stays fixed in the browser window between the two
clicks. You may have to scroll after the clicks to get to the buttons.
Click the button "Click when done calibrating". A spreadsheet
will open up. Click "Start taking data".
Start tracking the ball. Position the cursor over the
ball. When you click the left mouse button, the time and the y-coordinate
of the ball will be entered into the spreadsheet. You will automatically
step to the next frame of the video clip. Make sure the video frame stays
fixed in the browser window while you take data.
Repeat for each frame in the video clip until the ball reaches the bottom end
of the meter stick. Then click "Stop Taking Data". Highlight and copy your table. Open Microsoft Excel, and paste the table
into an Excel spreadsheet. Depending on your browser, you may have to use
"Paste" (Edge) or "Paste Special, Unicode Text" (Chrome). Your spreadsheet will have two columns, time (s)
and y (m). If you followed the instructions above, the the y-axis points
up.
Produce a graph of position versus time.
Label the axes.
Describe your graph. Does it resemble a
straight line? If not, what does it look like?
Was the ball moving with constant velocity?
How can you tell?
Let us find the velocity of the ball as a function of time. We find v_{y}
= ∆y/∆t by dividing the difference in successive position by the difference in
the times the ball was at those positions.
In your spreadsheet type type v (m/s) into cell C1.
Type 0 into cell C2. The ball starts from rest, its initial
velocity is zero
Type =(B3-B2)/(A3-A2) into cell C3. Copy the formula into the
other cells of column C, down to the second-to-last cell.
Produce a graph of velocity versus time.
Label the axes.
Describe your graph. Does it resemble a
straight line? If not, what does it look like?
Right-click your data in the velocity versus time graph and choose
"Add Trendline". Choose "Type, Linear" and "Options, Display equation on
chart". An equation y = ax + b will appear on your graph, where the number a is
the slope and the number b is the y-intercept. What is
the physical meaning of the slope of the velocity versus time graph, if the
graph is a straight line?
Paste your velocity versus time graph (with
trendline) into your log.
What value do you obtain for the acceleration of
the ball? How does your experimental
value of the magnitude of the acceleration compare to the accepted value of
the magnitude of the acceleration of a free-falling object?
Reminder: percent difference = 100%* |accepted value - experimental
value|/accepted value
What factors do you think may cause your
experimental value to be different from the accepted value? In other words,
what are some possible sources of error?
For motion with constant acceleration we expect that y changes as a function
of time as y = y_{0} + v_{0}t + ½at^{2}, where a is the
acceleration. For an object accelerating at a constant rate g we have y = y_{0} + v_{0}t
+ ½gt^{2},
so y as a function of t is a polynomial of order 2 (a section of a parabola).
We can reduce numerical errors in finding the acceleration of the ball by
fitting our position versus time data directly with a polynomial of order 2.
Right-click the data in your position versus time graph and choose "Add Trendline".
Choose Polynomial, Order 2 and under options click "Display equation on
chart". An equation of the form y = b_{1}x^{2} + b_{2}
x + b_{3} will be displayed where b_{1}, b_{2}, and
b_{3} are numbers. Since we are plotting y versus t, the number b_{1} is the best
estimate for g/2 from the fit. Therefore the value of the acceleration
determined from the fit is g = 2b_{1}. Since our y-axis points
upward, we expect a to be close to g = -9.8 m/s^{2}.
Paste your position versus time graph (with
trendline) into your log.
Does the polynomial of order 2 fit the data well?
What value do you obtain for the acceleration of the ball from this fit?
Projectile motion
Experiment 2
A ball is moving in two dimensions under the influence of a constant
gravitational force.
It is hard to get detailed information about its motion without using external
measuring instruments. In this experiment, the instrument is a video
camera. You will analyze a video clip. The clip shows a ball being
thrown. You will determine the position of the ball in two
dimensions as a function of time by stepping through the video clip
frame-by-frame and by reading the time and the position coordinates of the ball
off each frame. You will construct a spreadsheet with columns for time and
position and use this spreadsheet to find the x and y component of the velocity
as a function of time.
Procedure:
To play the video clip or to step through it frame-by-frame click the "Begin"
button.
"Play" the video clip. When finished, "Step up" to frame 1.
In the setup window, choose to track both
coordinate of the object.
Click "Calibrate".
Click "Calibrate X".
The video clip contains
two meter sticks. Position the cursor over left end of the horizontal stick and click the left mouse button. Then position the cursor
over the right end of the horizontal stick and click the left
mouse button again. This will record the x-coordinates of the chosen
positions. Enter the distance between those positions into the text box in
units of meter. For the example positions, you would enter 1 into the
text box. Click "Done".
Make sure the video frame stays fixed in the browser window between the two
clicks. You may have to scroll after the clicks to get to the buttons.
Now click "Calibrate Y".
Position the cursor over bottom end of the vertical stick and click the left
mouse button. Then position the cursor over the top end of the vertical
stick and click the left mouse button again. This will record the
y-coordinates of the chosen positions. Enter the distance between those
positions into the text box in units of meter. For the example positions,
you would enter 1 into the text box. Click "Done".
Make sure the video frame stays fixed in the browser window between the two
clicks.
Click the button "Click when done calibrating". A spreadsheet
will open up. Click "Start taking data".
Start tracking the ball. Position the cursor over the
ball. When you click the left mouse button, the time and the x- and y-coordinates
of the ball will be entered into the spreadsheet. You will automatically
step to the next frame of the video clip. Make sure the video frame stays
fixed in the browser window while you take data. When the ball is caught, click "Stop Taking
Data".
Your table will have 3 columns, time (s), x( m), and y (m).
Open Microsoft Excel, and paste the table into an Excel spreadsheet.
Produce graphs of the x and y components of position versus time.
Label the axes.
Describe the graphs. Does one of the graphs resemble a
straight line? If yes, what does this tell you?
Right-click your data in the x(m) versus time graph and choose
"Add Trendline". Choose "Type, Linear" and "Options, Display equation on
chart". An equation y = ax + b will appear on your graph, where the
number a is the slope and the number b is the y-intercept. What is
the physical meaning of the slope of this graph?
Right-click the data in your y(m) versus time graph and choose "Add
Trendline". Choose Polynomial, Order 2 and under options click "Display equation on
chart". An equation of the form y = b_{1}x^{2} + b_{2}
x + b_{3} will be displayed where b_{1}, b_{2}, and
b_{3} are numbers. What do the
coefficients b_{1} and b_{2} tell you?
We can view the motion of a projectile as a superposition of two
independent motions. Describe those two motions.
Paste your graphs with trendlines into your log.
Measuring the coefficients of static friction
In this experiment you will measure the coefficient of static
friction for a wood block and a felt-covered block in contact with a metal
track. The surface of the track makes an angle θ with the horizontal.
For angles θ > θ_{max} the block will accelerate down the sloping
track.
You will determine the angle θ_{max} for which the maximum force of
static friction f_{s_max }= μ_{s}N = μ_{s}mg
cosθ_{max} just cancels the
component of the gravitational force f_{g} = mg sinθ_{max}
pointing down the track. You will then solve for the coefficient of static
friction μ_{s}.
Experiment 3
Find the coefficient of static friction.
Pictures of the track and the wood block are shown below. The block has
a mass of 112.4 g and one side of the block is covered with felt. A
horizontal and a vertical meter stick are taped to the wall behind the track and
can be used to calibrate the video clips.
You will examine two video clips. In each clip the angle the track
makes with the horizontal slowly increases. Stop the clip when the block
starts moving and step up or down frame-by-frame to get to the frame just before
the block first moves.
Measure the angle the track makes with the horizontal in this frame.
Use a protractor or use a ruler and measure the width and height of a triangle
and use tanθ = height/width. You can also use an PhyPhox app on your
phone to measure the angle. Determine μ_{s} for the wood block and the felt-covered block.
To play or step through the video clips frame-by-frame click the buttons
below.
Construct a table as shown below. Insert this table
into your log.
surface
θ_{max} (deg)
μ_{s}
wood
felt
Comment on your results. Explain how you measured the
angle θ_{max}.
Convert your log into a lab report.
Name: E-mail address:
Laboratory 3 Report
In one or two sentences, state the goal of this lab.
Make sure you completed the entire lab and answered all parts. Make
sure you show your work and inserted and properly labeled relevant tables
and plots.
Add a reflection at the end of your report in a short essay format.
Save your Word document (your name_lab3.docx), go to Canvas, Assignments, Lab
3, and submit your document.