PC1221 Fundamentals of Physics I
Uniformly Accelerated Motion
1
Purpose
• Investigate how the acceleration of a cart rolling down an inclined track is related to the
angle of incline.
• Determine the acceleration of the cart from measured displacement and elapsed times.
• Determine an experimental value for g (the acceleration due to gravity) by a least square
fit to the acceleration versus sin θ data (θ is the angle of incline). Compare with the
accepted value of g.
2
Equipment
• Dynamics cart and track
• Adjustable ladder
• Stopwatch
• Meter rule
3
Theory
When an object undergoes one-dimensional uniformly accelerated motion, its instantaneous
velocity increases linearly with time. If it is assumed that the initial velocity of the object is
zero at time t = 0, then its instantaneous velocity v at any later time t is given by
v = at
(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 instantaneous velocity
is zero at t = 0 and it is v at any later time t. Therefore, the average velocity v during the
time interval is
0+v
v
v=
=
(2)
2
2
Level 1 Laboratory
Page 1 of 4
Department of Physics
National University of Singapore
Uniformly Accelerated Motion
Page 2 of 4
The displacement x of the object during the time interval is given by
x = vt =
vt
2
(3)
Substituting equation (1) for v in equation (3) gives
x=
at2
2
(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 1: A cart on an inclined plane.
A cart shown in Figure 1 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, it is:
a = g sin θ
4
(5)
Experimental Procedure
P1. Set up the track as shown in Figure 1 with the adjustable ladder. Elevate the end of
the track by about 10 cm.
P2. Measure the distance between two ends of the track. Record this distance as L in the
Data Table 1.
P3. Choose two positions, P1 (top) and P2 (bottom), on the track as the initial and final
positions of the cart. Measure the distance between these positions and record it as d
in the Data Table 1.
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Department of Physics
National University of Singapore
Uniformly Accelerated Motion
Page 3 of 4
P4. Pull the cart up to the initial position P1 on the track where the cart will be released
from rest.
P5. Release the cart from rest and use the stopwatch to tome how long it takes the cart
to reach the final position P2 . The person who releases the cart should also operate
the stopwatch. Repeat this measurement EIGHT times (with different people doing the
timing). Record all the values in the Data Table 1.
P6. Lower the end of the track by 1 cm and repeat step P5. Use the same release position
and final position.
P7. Repeat step P5 for a total of SIX angles, lowering the end of the track by 1 cm for each
new angle.
5
Data Analysis
D1. Enter your data in the Data Table 1 into the Excel spreadsheet. For each height of the
track, calculate the angle of inclination θ and the value of sin θ in the spreadsheet.
D2. For each height of the track, calculate the average time t̄ for the cart to move from P1
to P2 in the spreadsheet.
D3. Use the distance traveled d and average time t̄ to calculate the acceleration of the cart
a for each height of the track in the spreadsheet.
D4. Perform a linear least squares fit to the data with acceleration a as the vertical axis and
sin θ as the horizontal axis. Determine the slope and intercept with the corresponding
uncertainties of the least squares fit to the data.
D5. Plot a graph of the acceleration a against sin θ in the spreadsheet. Also show on the
graph the straight line that was obtained by the linear least squares fit to the data.
D6. Determine the experimental value of the acceleration due to gravity g with the corresponding uncertainty.
D7. Look up for the accepted value of g at Singapore and record it in your laboratory report
with the corresponding source of information. Use percentage discrepancy to compare
your experimental value of g with the accepted value.
Hint: The percentage discrepancy is defined as
Percentage discrepancy =
Level 1 Laboratory
|Experimental value − Accepted value|
× 100%
Accepted value
Department of Physics
National University of Singapore
Uniformly Accelerated Motion
6
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Questions
Q1. Would friction tend to cause your experimental value for g to be greater or less than
the accepted value? In which direction is your uncertainty for the value for g? Could
fiction be the cause of your observed uncertainty?
Q2. Assume there was a systematic uncertainty caused by a constant reaction time in pressing the stopwatch, would it cause an uncertainty in the direction of the observed difference between the experimental value and the accepted value of g?
Q3. What is the maximum possible value of the acceleration for a cart released from rest on
a track elevated at one end? Describe the experimental setup in this case.
Level 1 Laboratory
Department of Physics
National University of Singapore