02 Me1 PDF
02 Me1 PDF
02 Me1 PDF
OBJECTIVES
2. To confirm the conditions for static equilibrium of concurrent forces in two dimensions.
INTRODUCTION
All real bodies, living and otherwise, experience forces in their physical environments. The force
of gravity acts on all bodies. Right now, gravity is pulling down on you, and your chair or
whatever is supporting you supplies an equal force in the upward direction, resulting in
equilibrium. Creatures which live in fluids experience a frictional drag force when they move in
that fluid. A clear understanding of forces is necessary in order to appreciate how various living
things are affected by the forces in their particular environments.
In this experiment, a static system of concurrent forces is set up, and the lines of action of the
forces are traced onto a sheet of paper. A scaled vector diagram is constructed in order to
show that the vector sum of the forces is zero. The force system is also analyzed by finding the
components of the individual forces along two mutually perpendicular axes, and showing that
the sums of the two sets of components are zero.
THEORY
Any number of vectors can be added head-to-tail in this way. Because of its shape, the figure is
often called a vector polygon.
Bx = B cos , and
ME 1-2
b) Addition of components: The x-component and y-component of the resultant of several
vectors are the sum of the x-components of the individual vectors and the sum of the
y-components of the same vectors, that is,
c) Resultant from components: Once the components are known, the magnitude of the
resultant of several vectors may be found from
R = [(Rx)2 + Ry)2]1/2
Ry
= tan 1
Rx
Note that a calculator does not necessarily give the correct value for from this equation.
Instead, it always displays a value between -90O and +90O. As a result, the correct direction
must be found by inspection from the algebraic signs of Rx and Ry. An example should help to
clarify this.
Example: Suppose Rx = -1N and Ry = 1N. The fraction Ry/Rx = 1N/-1N = -1. Taking the
inverse tangent of -1 gives -45O, which lies in the fourth quadrant. By inspection,
though, since Rx is negative and Ry is positive, the resultant must lie in the second
quadrant. Thus, the calculator has given an answer which is 180O off. Perhaps the
easiest way to see this is to draw a small set of axes and to sketch on it the angle given
by the calculator and the approximate direction given by the signs of Ry and Rx.
APPARATUS
Vertical force board, 2 C-clamps, set of 4 strings, 2 sets of hooked masses, plain paper,
masking tape, small mirror.
PROCEDURE
Notes: 1. While partners will set up the apparatus together, each person will make a tracing of
the force system on a separate sheet of paper and use it for the calculations.
2. Your results are dependent on the care with which the various points are
marked and the lines are drawn. Use a sharp pencil and draw fine lines.
ME 1-3
2. Using the strings and the masses supplied, set up a system of four forces (three over the
pulleys and one hanging downward) so that the knot is suspended in equilibrium near the centre
of the clear space on the force board. None of the forces should have a value which is trivial
(very small) compared to the others.
Displace the knot from its equilibrium position and release it. The force system should oscillate
freely, coming to rest at its original position. Tap (vibrate) the force board in order to minimize
the effects of static friction in the pulleys.
3. Slide a sheet of paper between the strings and the force board, and locate it so that a
reasonable length of each string lies over the paper. Adjust the forces if necessary. Tape the
paper in place. The paper does not need to be square with the force board.
4. Mark the position of each string with tiny circled dots at two widely-spaced points. Use the
mirror to eliminate errors due to parallax. Label each pair of marks with the value of the mass
on the string, and draw a short arrow to indicate the direction of each force.
5. Remove the sheet of paper from the force board and put your name on it. Complete the force
lines and give each force a label, e.g. F1, etc..
1. Prepare a table similar to the following in your lab notebook, and enter the values in your
table as you proceed. Leave space beneath the Fx and Fy columns for their sums.
F2
F3
F4
2. Calculate the weight of each of the masses (W = mg). Show one sample calculation. Use
9.81 m/s2 for the acceleration of gravity. Keep one extra figure in the weights and other
calculated values in order to minimize round-off errors.
ME 1-4
Graphical addition of vectors:
3. Draw an (arbitrary) x-axis, which passes through the intersection of your force lines. See
Figure 4 for guidance.
Measure and record the (counterclockwise) direction angles of the vectors on your sheet.
4. On a separate page in your notebook, draw a scale vector polygon of the forces. Label each
of the forces as F1, etc. State your scale, and use as large a scale as the page will permit. Use
a sharp pencil and work carefully.
5. Clearly label the resultant vector (it should be very small). Measure its length and convert the
length to force units. Measure the direction angle. Reference the angle to the positive x-axis in
the conventional way. State the magnitude and the direction of the resultant on your polygon.
6. Enter the angles from your drawing into the table. Find the values of the sines and cosines of
the angles. Note that the number of significant figures in a trigonometric function does not
depend on the size the angle. For example, 2O and 358O are both measured to the nearest
degree.
7. Calculate the x- and y-components of each of the forces. Use SI units. Show one sample.
8. Sum the x-components and the y-components at the bottom of the columns under the table.
Show both the unrounded and the rounded sums.
Are both of the rounded sums equal to zero? If they are not, find the percent difference
between the sums of the positive and negative components, without regard to sign.
The hooked masses used in this experiment may be as much as 1% above their listed values.
A percent difference of less than this amount is within the tolerance of the masses.
If your percent difference is not within the 1% tolerance allowed by the hooked masses, what
other experimental factors might account for it?
9. Calculate the magnitude and the direction of the resultant from the unrounded sums of the
components. Reference the angle to the positive x-axis in the conventional way.
ME 1-5