Lab Problem 5: Velocity and Force
Jenny Smith
July 14, 2012
Physics 1201W, Professor XXXX, TA YYYYYY
Introduction
As part of an investigation of dinosaur flight, we are building a model launch mechanism. If
successful, this model could be scaled up to launch models of flying dinosaurs to study their
aerodynamic properties. As an initial step, we modeled the launch mechanism with a cart
accelerated down a straight, level track by a string attached to a hanging mass as shown in
Figure 1. Our goal is to test our prediction of the final velocity of the cart as a function of the
hanging mass and the height of its release.
+x +y Figure 1: The apparatus used (not to scale).
is the mass of the cart, is the mass of the
hanging object, is the initial distance of the cart to the end of the track, is the initial height
of the hanging object above the floor, is the acceleration of both the cart and the hanging
object. Note:
. Also shown: coordinate system used.
Prediction
The final velocity can be calculated by analyzing forces using ∑
For this purpose, we
consider the forces on two objects, (1) the hanging mass and (2) the cart. Since they are
connected by a string, both objects have the same acceleration and velocity and travel the same
distance until the hanging object hits the floor. The free body diagram for each object is shown
in Figure 2 together with the forces for each object on a coordinate system.
+x a +x Wo +y +y is the weight of the cart, is the
Figure 2: Free body diagram of cart and hanging object.
is the weight of the
normal force of the track on the cart’s wheels, is the pull of the string,
hanging object. Also shown are the forces on a coordinate system.
If the cart’s wheels turn freely, we assume the frictional force is negligible. Calling the mass of
the hanging object m and the mass of the cart M, adding the forces for the hanging object gives
–
. Where is the pull of the string on the hanging object, is the gravitational
acceleration on Earth, and is its acceleration. Likewise, for the cart,
two equations, we can eliminate and solve for .
. With these
(1)
Note that the acceleration of both the cart and the hanging object is constant. Using the
definition of acceleration for the cart,
, we can solve for the distance it moves, , as a
function of its acceleration
(2)
where is the time for the cart to move a distance assuming the initial velocity of the cart is
zero. The velocity of the cart is related to its acceleration by
, giving
(3).
Combining equations (2) and (3) to eliminate t gives
2 (4).
Combining equations (1) and (4) to eliminate a gives
2
(5).
Since the cart and the hanging object are tied by a string the distance the cart moves, , is the
same as the distance the object falls, .
2
(6).
After the hanging object hits the floor, there is not horizontal force on the cart since we are
assuming friction is negligible. That means this maximum velocity will be the constant
velocity of the cart until it hits the end of the track.
Apparatus & Procedure
First the wheels of the cart were checked to make sure that they did turn freely to conform to
the frictionless assumption. The cart was placed on a straight, level, elevated track with a
pulley at one end as shown in Figure 1. A string was tied to the cart and run over a pulley. The
string connecting the cart and the hanging object was longer than the height of the object so that
the object would hit the ground and the cart would still be moving. At that point until the cart
hit the end of the track, there were no horizontal forces on the cart. Initially, the cart was held
and then released. The cart accelerated until the hanging object hit the ground and then it
continued with a constant velocity until it hit the end of the track. The height of the hanging
object was measured with a meter stick and the mass of the object with a laboratory balance.
The cart mass was also measured with the same balance.
We tried a range of masses for the hanging objects before taking any measurements. Between
50 and 150 grams gave a speed that was slow enough to at least 10 video frames in the time it
took between the object hitting the ground and the cart hitting the end of the track. Using
smaller masses made the friction in the cart’s wheels non-negligible as seen from deviations of
the position vs. time graph from a straight line. Larger masses made the pulley more difficult to
turn which violated our assumption of no pulley friction. Within this range of masses, we had
four different hanging objects to use.
The motion of the cart was recorded with a video camera aimed perpendicular to the track at its
center and placed 2.1 m from the track. The software, MotionLab, allowed us to click on the
position of the cart for each frame of the video. We used the wheel of the cart as the position
reference, not the falling mass since it was generally centered in the frame of the video. Since
the video is taken at a constant frame rate, this gives the position of the cart as a function of
time. From this information, the software plots a position vs. time graph and a velocity vs. time
graph. We then matched a function to each graph to quantitatively describe the motion.
Examples of two graphs of the motion of the cart after the hanging object hits the ground but
before the cart hits the end of the track are given in Figure 3. Note that the position vs. time
graph is a straight line which means the velocity is constant as is shown by the velocity vs. time
graph.
Figure 3: Left hand graph is position of the cart (in cm) vs. time in seconds after hanging
weight hits the ground but before the cart hits the end of the track. The red points are the data.
The purple line is my best match of a function to the data. This straight line has a slope of 122
cm/s. The right hand graph is the velocity of the cart vs. time for the same time interval. The
blue line is my best match of a function to the data and is a constant.
Data
To determine how well the prediction matched the measurement, we decided to make the
measurement for hanging objects of four different masses when released from the same height.
The results are given in Table 1. The uncertainty in the prediction comes primarily from the
measurement of the height of the object with some contribution from the measurement of the
mass. The measurement could be made two different ways by determining: (1) the slope of the
position vs. time graph or (2) the constant value of the velocity vs. time graph. After seeing
what variation the line could have and still adequately represent the points, we decided that
technique (1) had smaller uncertainty and was used.
in
grams
in
meters
Prediction of
in m/s
50
70
100
150
0.46
0.41
0.47
0.48
1.24
1.34
1.64
1.94
Prediction
uncertainty in
m/s
±0.01
±0.01
±0.01
±0.01
Measurement of
in m/s
Measurement
uncertainty in m/s
1.22
1.41
1.65
1.90
±0.07
±0.05
±0.05
±0.03
Table 1: is the mass of the hanging object and is its release height. The prediction is
calculated from equation (6). The measurement is from the slope of the straight line that best
matches the position vs time graph. The measurement uncertainty of mass was estimated to be
1 gram and that of the release heights, 2mm.
Analysis
The experimental measurements and predictions of the final velocity are given in Table 1 in the
Data section. The uncertainty in both the prediction and measurement were calculated by the
"worst case" technique. The acceleration due to gravity, ,was assumed to be precisely 9.80
m/s2 for this calculation. The average absolute value of the difference between the prediction
and the measurement was 0.035 m/s which is well within the average uncertainty of 0.05 m/s.
There are many potential sources of uncertainty that were minimized by our choice of
equipment and procedure. These included: the friction of the cart wheels, the friction in the
pulley, the deviation of the track from horizontal, the distortion caused by the camera, the mass
of the string and the pulley, and the actual value of g in Minneapolis. All of these were judged
to be smaller than our measurement uncertainty.
Conclusion
We have tested model of a proposed launching mechanism to determine how well a simple
theoretical calculation would predict its final velocity. We found that within experimental
uncertainties that the measured final velocity agreed with that calculated by analyzing the
forces and using a straight forward application of Newton’s 2nd Law. This approach is useful
when other effects, given in the analysis section, are negligible.