Experiment No. 1 bsc.2nd semester
1.
Determination
and Verification of the Time Period of a Cantilever
Object
To
determine and verify the time period of oscillation of a cantilever beam whith
a known mass attached at its free end.
Apparatus Required
1.
Cantilever beam (e.g a steel or
aluminum ruler)
2.
Clamp and stand to fix the beam.
3.
Set of known masses.
4.
Stopwatch
5.
Measuring scale or ruler
6.
Vernier caliper or micro meter screw
gauge.
7.
Balance (for measuring mass)
Theory
When
a mass is attached to the free end of a cantilever beam and displaced slightly
it undergoes simple harmonic motion. The time period T of oscillation is given
by –
T
= 2π √ML³/√3EI
For a rectangular cross – section .
I
= bd³/12
Where b
= width of the beam
.d
= thickness (depth) of the beam.
Procedure
1.
Setup secure one end of the beam
finely using the drop and stand ensuring it acts as a cantilever.
2.
Measurement measure the length L from the
fixed end to the point where the mass will be attached.
3.
Cross – sectional dimensions use the
vernier caliper or micrometer to measure the with b and thickness d of the
beam.
4.
Mass attachment- attach a known mass M
to the free end of the beam
5.
Oscillation – displace the mass
slightly downward and release to allow oscillation.
6.
Timing – use the stopwatch to measure
the time taken for a certain number of oscillation “(e.g. 20 oscillation) to
minimize error.
7.
Repeat – repeat the timing for
multiple trials to obtain an average value.
8.
Variation – repeat the experiment with
different masses and or lengths to observe the effect on the time period.
Observation
|
S. No. |
Mass M (Kg) |
Length L (m) |
Width b (m) |
Thickness d (m) |
Time for 20 oscillations (s) |
Time period T (s) |
|
1. |
0.100 |
0.30 |
0.025 |
0.003 |
16.2 |
0.81 |
|
2. |
0.150 |
0.30 |
0.025 |
0.003 |
19.0 |
0.95 |
|
3. |
0.200 |
0.30 |
0.0025 |
0.003 |
21.5 |
1.06 |
|
4. |
0.250 |
0.30 |
0.025 |
0.003 |
23.8 |
1.19 |
|
5. |
0.300 |
0.30 |
0.025 |
0.003 |
26.0 |
1.30 |
|
|
|
|
|
|
|
|
Calculations
1.
Moment of Inertia I
I
= bd³/12 = 0.025 x (0.003)³/12 = 5.625 x 10⁻ⁱ⁰ m⁴
2.
Theoretical time Period T-
T
= 2π √ML³/√3EI
Assuming
–
a.
Young’s modulus for steel E = 2 x 10ᴵᴵ
Pa
b.
Length L = 0.30 m
c.
Moment of inertia I = 5.625 x 10ᴵ⁰ m⁴
For Trial 1 (M = 0.100 kg)
3. Percentage , calculate for other trials.
Percentage error
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Result -
The
experimental time periods closely match the theoretical predictions with
minimum percentage error validating the theoretical madel for the cantilever’s
oscillations.
Calculation
The
experiment successfully demonstrates the relationship between the mass attached
to a cantilever and its oscillation period . the close agreement between
experimental and theoretical values conform the validty of the theoretical
formula used.
Precautions
a.
Ensure the beams rigidly clamed to
prevent unwanted movements.
b.
Measure dimensions accurately using
appropriate instruments.
c.
Avoid large displacements to maintain
simple harmonic motion conditions.
d. Use consistent method for timing oscillations to reduce human error.
Note.
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