Download Acceleration of a Cart

Circular Motion
LBS 164L
Purpose
In this experiment you will investigate the force required to keep an object moving in a
circle.
Theory
In this experiment, a spherical mass is suspended from a string as shown in Figure 1. The
mass is pulled back to a given height, h, above the original height and released. The mass
will begin to swing back and forth.
PASC O
L
L
m
m
Figure 1
h
Let’s consider two points during this
motion, the highest point and the lowest point. The
conservation of energy tells us that
 K top U top  K bottom  U bottom.
We know that at the top
 K top  0 .
At the bottom we define
 U bottom  0 .
Energy conservation then tells us
 U top  Kbottom
which can we rewritten as
1
 mgh  mv 2 .
2
We can then find the velocity at the bottom to be
 v  2gh
When the mass is moving at the bottom of its trajectory, the forces acting on it are
the tension on the string and the force of gravity. The resulting acceleration is due to the
fact that the string is forcing the ball to travel in a circle. The free-body diagram shown
in Figure 2 describes this situation.
The sum of the forces in the y direction equals the mass times the
centripetal acceleration because the string does not stretch or break. So we
can write
  Fy  T  mg  ma centripetal .
We know that
v2
 a centripetal 
L
so we can write the tension on the string at the lowest point as
Figure 2
mv 2
 T  mg 
.
L
Circular Motion Lab Write-up
Page 1
In this experiment you will experimentally verify the following concepts:
 The velocity of the object at the bottom of the trajectory does not depend on
the mass of the object.
 The velocity of the object at the bottom of the trajectory depends on the height
to which it is raised.
 The tension on the string at the bottom of the trajectory depends on the mass
of the object and velocity of the object. The extra tension beyond the weight
of the object is due to the circular motion of the object.
Equipment Needed
Macintosh computer
PASCO CI-6560 Signal Interface II
PASCO CI-6521A Motion Sensor
PASCO Force Sensor
string
spherical weights
ring stand and clamps
Science Workshop
KaleidaGraph
Microsoft Word
Microsoft Excel
Procedure
The equipment you will need has been assembled and tested. To make sure that
everything is in order, carry out the following checks:
 Make sure the Signal Interface is turned on,
 Make sure that the Motion Sensor is connected correctly to the Signal Interface (cable
with the yellow band into Digital Channel 1 and the cable with the black band into
Digital Channel 2),
 Make sure that the Force Sensor is plugged into Analog Channel A on the Signal
Interface.
Set up the equipment as shown in Figure 3.
Force Sensor
Place the Motion Sensor such that it is
located about 70 cm from the hanging ball and is at
the same height as the ball.
Prepare the computer to record data. Open
the Science Workshop document “Circular
Motion”.
Motion Sensor
Figure 3
Circular Motion Lab Write-up
Data Recording
The two classes of information that we will
record are the horizontal velocity versus time and
the vertical force versus time. We will do these
measurements for two different heights and two
different masses. The vertical force is measured to
Page 2
v Dat a
1.5
v (m/sec)
1
v (m/sec)
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
Time (sec)
Figure 4
2.5
3
3.5
4
be negative and is the opposite of the
tension on the string. The object of each
measurement is to determine the
horizontal velocity and vertical force when
the mass is at the lowest point of its
trajectory. To accomplish this task,
several oscillations of the ball will be
recorded and the appropriate values taken
from the graphs. After each of the four
measurements, you should save a text file
from the tables for the velocity and force
measurements. Each measurement
includes at least three oscillations of the
ball. Record the length of the string from
swinging point to center of ball. Close
Science Workshop.
F Dat a
-0.3
Force (Newtons)
For ce (N ewtons)
Open KaleidaGraph and open all
-0.35
eight data files, skipping the first two lines
-0.4
as usual. Make eight plots based on these
data. In the velocity plots, the highest
-0.45
velocity corresponds to the horizontal
velocity at the lowest point. Using the line
-0.5
drawing tool of KaleidaGraph, draw a line
-0.55
through the maximum velocities as
illustrated in Figure 4.
-0.6
In a similar fashion, determine the
0
0.5
1
1.5
2
2.5
3
3.5
4
vertical force at the lowest point. In the
Time (sec)
case of the force, the vertical force at the
Figure 5
bottom of the trajectory corresponds to the
largest negative (downward) force
measured. In this case the line should be drawn through the lower values of the force
taking care to average the noise in the measurement. A typical example is shown in
Figure 5.
Measurements
Dependence on Height
 Measure the velocity and force at the bottom of the trajectory for two different
initial heights. Make sure that the angle of the ball with respect to the vertical
is not larger than about 20˚.
Dependence on Mass
 Measure the velocity and force at the bottom of the trajectory for two different
masses using the same initial height. Again make sure that the angle of the
ball with respect to the vertical is not larger than about 20˚. The two different
masses are accomplished using a golf ball and a golf ball with added weight.
Calculations
 Record the results of four measurements in an Excel spreadsheet. Calculate
the horizontal velocity and the tension on the string at the lowest point in the
trajectory based on your measured values for h, m, and L. Use the spreadsheet
template called “Circular Motion Spreadsheet” as shown in Figure 6.
Circular Motion Lab Write-up
Page 3
g:
L:
Mass (kg)
0.045
9.810
0.630
h (m)
v calc. (m/s)
0.060
1.085
v meas. (m/s)
T calc. (N)
1.050
0.526
T meas. (N)
0.540
v dif f .
1.67%
T dif f . (N)
2.68%
Figure 6
Questions
1. Did the measured velocity at the bottom of the trajectory depend on the height to
which the ball was raised?
2. Did the measured velocity at the bottom of the trajectory depend on the mass of the
ball?
3. Did the measured tension at the lowest point depend on the height to which the mass
was raised?
4. Did the measured tension at the lowest point depend on the mass of the ball?
5. Did the velocity at the lowest point calculated using m and h agree with the velocity
measured with the motion sensor?
6. Did the tension at the lowest point calculated using m and h agree with the tension
measured with the force transducer?
Circular Motion Lab Write-up
Page 4