__Experiment 2__

Rotational Equilibrium

Supports Reactions of a Loaded Beam

Objective:

1) To experimentally verify the equations for the equilibrium of non-concurrent forces by measuring the supports reactions of a loaded beam

Equipment:

A uniform metal beam, two triangular supports, two flat-top scales, weights, and a calculator

Theory:

The conditions for the static equilibrium of a loaded beam
are**:**

ΣF_{x}
= 0 ;
ΣF_{y}
= 0 ;
ΣT
= 0.

For a horizontal
beam that is supported at two points and loaded as shown, the external forces
acting on it are all vertical**.** Since there are no horizontal forces
acting on it, the condition ΣF_{x}
= 0 is automatically satisfied**.**
By applying the equations**:**
ΣF_{y}
= 0 and
ΣT
= 0, two unknown reactions at its two
supports can be calculated and then compared with their corresponding measured
values**.** This verifies the validity of the net torque equilibrium
equation ΣT
= 0**.** Note that although no
rotation takes place in this experiment**;** however**,** since the sum of
torques caused by forces is set equal to zero for static equilibrium**,** we say
that rotational equilibrium is achieved**.** If a rotating object is
rotating at a constant angular velocity, its angular
acceleration α = 0.
This makes the net torque acting on it to be zero that means
ΣT
= Iα = 0. If an object is not
rotating and is at rest, the equation
ΣT
= Iα = 0 still holds true because α = 0
due to no rotation**.**

The point about which the sum of torques or moments
is to be calculated is arbitrary**.** Any point may be selected
for this purpose**.**
The point may be chosen on the beam or out of
it**.** The final values that one calculates for the reactions will come out
the same regardless of the point chosen**.**

__
Procedure:__

Note: Since this
experiment involves weights of up to 5-kg, it is recommended that the
experimental set up be at the center of the table and far from the edges** **
to avoid weights accidentally fall on the floor**.** Although all
students must have covered-toe-shoes on, those with weaker shoes must
specifically keep distance from the table edges**.**

The beam to be used is
divided into 8 equal sections by 8 lines, as shown**.** The supports
are triangular prisms, as shown**.** When placed on the table, each
support has a relatively sharp top edge for the beam to be placed on top of it**.**
This guarantees the measurement of the position of the reaction forces on the
beam with good precision**.**

For the purpose of force measurement, two identical
flat-top scales must be used**.** On the top of each scale, one
triangular support must be placed and then zeroed**.** This way, the
weights of the supports will be excluded from calculations**.** The
beam should then be placed on the top edges of the supports and its position
adjusted** **as shown**.**

Following are the specific steps to be taken**:**

__Part 1__

1) Determine the weight of
the aluminum beam by using an appropriate scale provided for this experiment**.**

2) Place the supports at B
and H by adjusting the positions of the scale, supports, and beam**.**

3) Place a 5-kilogram weight centered at B**.** Read
the
reaction forces (from the scales) and record them in the table provided under
the Data section**.**

4) Move the 5-kg weight to
C**.** Record the new reactions**.**

5) Repeat
Step 4 with
the 5 kg weight moved to positions D, E, F, G and H respectively**.**

6) Calculate the expected
or accepted values of the reactions for each case by applying the two equilibrium equations
discussed above**.** Determine a %error on each reaction in each
case**.**

7) Plot the experimental
values of reaction at B versus the location of 5 kg weight on the beam**.**

__Part 2__

1) Place the supports at E
and H**.**

2) Place a 5-kg weight at
G**.** Read scale reactions and record them in a similar table**.**

3) Let the 5-kg weight
remain at G**.** Place a 1-kg weight at D**.** Record
the new reactions**.**

4) Repeat
Step 3 with the
one-kilogram weight moved to the C and B positions, respectively**.**

5) Calculate the
accepted values of reactions and determine a %error on each reaction as you did
in Part 1**.
**

6) Plot the analytical
reactions at location H versus the location of the 1- kg weight on the
beam**.**

From the graph
in Step 6,
determine the location of the 1-kg weight when the reaction at point H
is zero**.** (It will be off the beam and therefore it is somewhat hypothetical)**.**

**
Data:**

**
Given: **

g = 9**.**81 m/s^{2}**.**

**Measured:**

** **

Part 1 | Case | Load
(kg) |
Measured Reactions (N) | Calculated Reactions (N) | %error on R_{B} |
%error on R_{H} |
||

R_{B} |
R_{H} |
R_{B} |
R_{H} |
|||||

1 | 5 at A | |||||||

2 | 5 at B | |||||||

3 | 5 at C | |||||||

4 | 5 at D | |||||||

5 | 5 at E | |||||||

6 | 5 at F | |||||||

7 | 5 at G | |||||||

8 | 5 at H | |||||||

Note: Observe significant figures | ||||||||

Part 2 | Case | Load
(kg) |
Measured Reactions (N) | Calculated Reactions (N) | %error on R_{E} |
%error on R_{H} |
||

R_{E} |
R_{H} |
R_{E} |
R_{H} |
|||||

1 | 5kg at G | |||||||

2 | 5 at G, 1 at D | |||||||

3 | 5 at G, 1 at C | |||||||

4 | 5 at G, 1 at B | |||||||

Note: Observe significant figures |

__Comparison
of the results:__

Provide the percent error
formula used and the calculated percent errors in each case**.**

__Conclusion:__

State your conclusions of
the experiment**.**

__
Discussion:__

Provide a discussion if
necessary**.**