Static Equilibrium of a Loaded Beam
Review 106-Statics.pdf and Giancoli sections 10.5 and 12.1-3.
Introduction
If an object remains stationary (i.e. static) the fact that its center of mass does not move
requires that the net force acting on it is zero. Similarly, the fact that it does not rotate
requires that the net torque acting on it is zero. The equations expressing these two concepts
are:
X
F~i = 0
X
~τi = 0
(1)
(2)
where F~i is a force acting on the object and ~τi is a torque acting on the object. In this lab
you will make measurements of the forces on a beam with a load and determine if your data
supports this. The estimate of measurement error in this lab requires careful consideration.
Inaccurate estimates of error may lead to incorrect conclusions. Typical measurement device
errors are ‘half of the smallest division’. However, in this lab there are many factors which
may make your measurements less precise than the ‘measuring device error’. These include:
• parallax (which will cause a variation in the position of markings made on the board),
• the thickness of the lines drawn with the marker, and
• friction in the spring balances.
You should try and make realistic estimates of these errors as you proceed through the
experiment.
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Equipment Supplied
You are supplied with the following: a magnetic white board which stands vertically, a beam
with wheels on one end, weights, spring balances and a pulley to support the upper end of
the system.
Procedure and Analysis
Arrange the beam of mass m, as shown in the figure, with the wheels resting on the bench
and against the board support block. The lower end of the beam with the wheels is point
A. Fix the pulley to the spring balance and the other end of the spring balance to the top
end of the beam. The spring balance provides a measure of the cable tension FT . Attach a
weight (M g) to the upper hook on the beam as shown.
pulley
board
FT
spring
balance
θ2
FyA
wheels
support
block
θ1
support
block
A FxA
mg
Mg
FIG. 1: Schematic picture of the apparatus.
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Note both the mass m of the beam, and the mass mw of the beam with the wheels (written
on the beam) and measure the angles θ1 and θ2 when it is set up as in the figure. Read
the tension FT from the spring balance. You can also determine the x and y forces on the
beam due to the table and vertical support at point A. Determine the force FxA by attaching
a spring balance to the lower end of the beam and pulling horizontally with just enough
force to cause the end of the beam to come off of the side wall. Determine the force FyA
by attaching a spring balance to the lower end of the beam and pulling vertically with just
enough force to cause the end of the beam to lift off of the table. The information you have
P
just recorded will allow you to calculate the net forces (
P
(
Fx and
P
Fy ) and the net torque
~τ ) on the beam. Record the necessary data in your lab book using a table like the one
below. Also include a sketch in your lab book with all variables labeled.
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Short Report (To be handed in):
Name:
Partners Name:
Student ID:
Date:
1. The following table:
VALUE
UNCERTAINTY
FT , tension in spring scale (N)
θ1 , angle between beam and horizontal (deg.)
θ2 , angle between beam and spring scale (deg.)
M , mass of load (kg)
m, mass of beam (kg)
mw , mass of beam with wheel (kg)
FxA , horizontal force on base of beam (N)
FyA , vertical force on base of beam (N)
2. Your values for the sum of forces (ΣFy and ΣFx ) and the sum of the torques about
the axle of the wheels (Στ ), with their errors. Include one sample torque calculation
and show the associated uncertainty calculation.
3. In the force calculations you need the mass of the beam with the wheels but in the
torque calculations you should use the mass of the beam without the wheels, explain
why?
4. Explain how you estimated uncertainty in the force and angle measurements. Were
there any uncertainties that may have been incorrectly estimated due to simplifying
assumptions? List these and explain why they were difficult to estimate and how the
estimates could be improved.
5. Your values of ΣFy , ΣFx , and Στ should all be zero. Are they zero within experimental
uncertainty? Explain why or why not.