Centripetal Acceleration
A pendulum consists of a weight (known in this context as a bob) on the end of a string (or rigid
rod). The mass experiences two forces: the weight of the bob (directed vertically downward) and
the tension (which acts along the string).
If two forces are not collinear, they cannot possibly cancel out. For a swinging pendulum there
is a net force and hence the pendulum bob accelerates. Typically one uses a radial-tangential
coordinate system (shown in the figure above). At an arbitrary point in the bob's swing, the net
force has both radial and tangential components. As a result there are two types of acceleration:
centripetal acceleration (which is directed radially) and angular acceleration (which is directed
tangentially).
When the pendulum bob is at the lowest point in its swing (when the radial and vertical
directions coincide), the forces are collinear, however, the forces have different magnitudes.
Which is bigger? The net force is purely radial and hence the acceleration purely
centripetal. The magnitude of centripetal acceleration is given by
acentripetal = v2/r
Thus at the bottom of the swing, the net force (Tension - Weight) is responsible for the
centripetal acceleration.
Tension - Weight = m acentripetal
A swinging pendulum is never in equilibrium (i.e. there is always a net force).
In this lab we will use a Force Sensor to measure the tension and will assume conservation of
energy to calculate a speed.
Measurements and Analysis

Measure the diameter and height of a cylindrical weight and record its mass. Use a brass
or aluminum cylinder.
Cylinder Properties
Diameter ( ) Height (
) Mass ( )

Using a tall ringstand, clamp, Force Sensor, cylindrical weight and string, make a
pendulum set up like the one shown below.

In Data Studio, click on the Analog Channel A icon. Choose Force Sensor from the
menu.
Click on the Sampling Options button (on the left) and change the Periodic Samples to
100 Hz.




Hold the cylinder to remove the tension from the string and then Tare the Force Sensor.
While the cylinder is not swinging, record its weight (the mean reading of the Force
sensor) in the table. (Is this value consistent with the mass reading taken above?)
Note where the center of mass of the bob is when the mass is hanging straight down, then
pull the mass up and over (at some angle to the vertical) and note the new height of the
bob. Then release the bob (so its initial velocity is zero) and record a few swing’s worth
of data. Record the change in height in the table below.
Pendulum Length =
Weight ( )
Tension at swing Net force at swing Change in Height
Velocity ( )
bottom ( )
bottom ( )
of Bob ( )

Drag the graph icon into your Force run. Click the button that expands the graph to fill
the region available.

Click on the cross-hairs button, to estimate a typical force at the bottom of the
pendulum's swing. Recall the pendulum's tension is maximal at this point. Try to place
the cross-hair point so that vertically it is above some of the bottom points and below
other bottom points. The x and y coordinates of the cross-hair point can be seen in the x
and y axis labels.
We will use the conservation of energy to calculate the velocities. The conservation of
energy in this context says that the gravitational potential energy given by mgh where h
is the change in height you recorded above is converted into kinetic given by mv2/2.
Basically this is just the velocity the bob would have if it just fell.





Repeat the steps above starting with the taring of the Force Sensor and measuring the
weight (always checking for consistency with previous readings). But each time you start
the pendulum, swing it from a different height. Make measurements for four different
heights (yielding various velocities - but avoid small velocities). Try to get as large a
range of velocities as possible, i.e. have a small angle and a large angle in your range of
angles. (The Force sensors tend to drift, this is why we want to tare them and re-measure
the weight each time.)
Calculate the net force on the pendulum bob when it is at the bottom of its swing
(Tension at swing bottom - Weight).
Plot this Net force versus velocity. Fit it to a power law.
Compare your power to the theoretical value. (See centripetal acceleration above.)
Fit
Theory
Fit
Theory Percent
Percent
Coefficient Coefficient
Power Power Difference
Difference
( )
( )


Extract the coefficient from your fit, enter it above and complete the table.
Change the length of the pendulum and repeat the measurements and analysis above.
Remember that the length is measured from the center of the bob.
Length =
Weight ( )
Tension at swing Net force at swing
Velocity ( )
bottom ( )
bottom ( )
Fit
Theory
Fit
Theory Percent
Percent
Coefficient Coefficient
Power Power Difference
Difference
( )
( )

Repeat with another pendulum length. Try to obtain a large range of pendulum lengths (a
short, a medium and a long).
Length =
Weight ( )
Tension at swing Net force at swing
Velocity ( )
bottom ( )
bottom ( )
Fit
Theory
Fit
Theory Percent
Percent
Coefficient Coefficient
Power Power Difference
Difference
( )
( )

Using the formula from your fits, determine the net force at a velocity of 1 m/s for each
of your lengths.
Pendulum length ( )
Interpolated net force
for v = 1 m/s ( )