L A B O R A T O R Y
Physics Laboratory Manual M Loyd
4
Uniformly Accelerated
Motion
OBJECTIVES
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□
□
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Investigate how the displacement of a ball down the inclined plane is related to the elapsed
time.
Determine the acceleration of the cart from an analysis of the displacement versus time data.
Determine an experimental value for g (the acceleration due to gravity) by interpretation of the
cart's acceleration as a component of g. Compare with the accepted value of g.
Apply rules for propagation of errors in experiments and in statistical analysis of a linear
regression model.
EQUIPMENT LIST
• Inclined plane
• Laboratory timer, stopwatch, or photogates
• Block to raise the air track to form an inclined plane
• Labquest and computer
THEORY
When an object undergoes one-dimensional uniformly accelerated motion its velocity increases linearly
with time. If it is assumed that the initial velocity of the object is zero at time t = 0, then its velocity v at
any later time t is given by
v = at
(Eq. 1)
where a is the acceleration which is constant in magnitude and direction.
Consider a time interval between t = 0 and any later time t. The average velocity v during the time
interval is
(Eq. 2)
The displacement x of the object during the time interval t is given by
(Eq.3)
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Physics Laboratory Manual
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Figure 4-1 Graphs of displacement versus time and displacement versus the square of the time for
uniformly accelerated motion. The displacement is linear with time squared;
Figure 4-2 Components of g, the acceleration due to gravity on an air
track.
Substituting Equation 1 for v in Equation 3 gives
(Eq-4)
Equation 4 states that if an object is released from rest its displacement is directly proportional to the square of the elapsed
time. Figure 4-1 shows graphs of both x versus t and X versus t2 for uniformly accelerated motion.
A cart shown in Figure 4-2 is placed on an air track that is raised at one end to form an inclined plane with an angle of
inclination of Θ. The acceleration due to gravity points directly downward but can be resolved into components that are:
perpendicular and parallel to the plane. The cart moves down the inclined plane with acceleration a, the component of the
acceleration due to gravity g acting down the plane. In equation form this is:
(Eq. 5)
E X P E R I M E N T A L
1.
P R O C E D U R E
Place block(s) approximately 10-15 cm in height under the support at one end of the inclined plane. Set up a track
of meter sticks for your ball to roll down the plane.
Laboratory 4 ■ Uniformly Accelerated Motion 45
Measure the height of the plane at the bottom of the ramp. This will be your reference height h0. Measure the height of the
plane at the top( or some other location) of the ramp. This will be the height h1 . Measure the length of the ramp (d(m)).
Have two other people do the same measurement to provide statistical confidence. Do not announce your value to
influence any measurement. Write down the numbers for h1 – h0= h(m) each part of data table.
2. Set up a photogate at the bottom and a second one 20 cm up the ramp. Determine the scale limited error in this
distance measurement and write it in the data table. This might be larger than you expect because it will be difficult to
measure the precise placement of the photogate sensor. Connect the lower photogate to DIGI 2 and the upper gate to
DIGI 1 on the labquest. Open an experiment file in the “Probes and Sensors” folder called “Pulse Timer-Two Gates.
After pressing collect, you have approximately 100 seconds to complete the five trials. Have one member of the group
carefully release the ball from rest just behind the interior sensor of the photogate. The program should plot the time
from Gate 1 to Gate 2. Record the five times in the data table.
3. Repeat for 40cm, 60cm, 80cm, and 100cm. Record all values in the Data Table.
C A L C U L A T I O N S
Keep only one digit in all standard errors and keep decimal places in all means and standard deviation that coincide
with the decimal -place of the standard error. You may use your calculator for these calculations ( 1 variable stats).
1. Calculate the mean time
distance.
t
, the standard deviation, and the standard error αt for the five trials of the time at each
2. Calculate the square of each of the mean times ( t )2 for the five distances.
3. Perform a linear least squares fit to the data with x as the vertical axis and t2 as the horizontal axis. Plot both x and y
error bars. Determine the values of the slope, correlation coefficient r, and the RMSE. From the slope obtained in the
least squares fit, calculate the acceleration a of the cart where a=2(slope) according to Equation 4.
4. Calculate the value of sin θ= h / d using, the values of
appendix 2 for the error!!!
h and d . Use the rules of division for propagation of error from
5. Use Equation 5 to calculate an experimental value for the acceleration due to gravity (gexp) from the measured values of
a and sin θ.
6. Using the precision in the acceleration from the graph are used to calculate the experimental value of g, they determine
the number of significant figures in your value of g. Use this precision when stating this value in the abstract.
7. Calculate the percentage error in your value of gexp compared to the accepted value of 9.80 m/sec2.
G R A P H S
1. Construct a. graph of the x versus ( t )2 data with x as the vertical axis and ( t )2 as the horizontal axis. Show on the
graph the straight line from the linear least squares fit.
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Name
Date
Lab Partners
Data Table
1st person
variable
h0 =
2nd person
3rd person
m
Mean
Std dev
Std error
Xxxxxxxxx
Xxxxxxxxx
xxxxxxxxx
h1 – h0= h(m)
d(m)
x
+/-____(m)
t1(s)
t2(s)
t3(s)
t4(s)
t5(s)
Calculations Tables
Error Propagation of sin θ( no u nit s! )
sin θ = _ _ _ _ _ _ + / - _ _ _ _ _
1st
2nd
3rd
4th
5th
x(m)
t
(s)
σn-1=
αt=
( t )2 (s2)
αt2=
Slope =
+/-
m/s2
a=
+/-
m/s2
g exp =
+/-
% error=
r=
RSME=
m/s2
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Questions
1. Would friction tend to cause your experimental value for g to be greater or less than 9.80 m/sec2 ? In which direction
is your error for the value for g? Could friction be the cause of your observed error? Could there be something else?
State your reasoning.
2. Show the calculations. Using the error for x(m), apply the rules of propagation to determine the precision in the
acceleration with its error for the 1st height. Since scale limited errors are involved, it is not appropriate to use
appendix2.
a= 2x/t2
a= ________+/- _______
3.
Show the calculations. Using the slope from your graph, calculate the acceleration. Use the error in the slope
given for precision and determine the error on a. This should be more precise than #2. It is important to remember
to use graph properties of the entire data set rather than individual data points.
d=(a/2)t2 (a/2)=________+/- _________m/s2
a= ________+/- _______
4.
Show the calculations. Use the rules of propagation to determine the precision of your value of gexp from the
errors of the acceleration (#3) and sinΘ. Appendix 2 is appropriate here.
gexp= ______________ +/- __________
5.
Perform the Standard Equivalency Test (SET) for gexp. State your response in the proper form as described in
Appendix 2.