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```PHY211 LAB 10
Simple Harmonic Motion
Abstract
The purpose for this lab is to demonstrate the effects of mass, length, and
angle upon a pendulum, and to compare the calculated gravitational effect
to actual gravity effect. Through a series of weights of 50 -150 g and m
lengths of the same numeric, the average time was calculated to find the
period of a 10-cycle swing. By changing the angle of the beginning swing,
this displacement will also be used to calculate the e ffect of starting
momentum on the mass.
Sanabria, Helen (PPCC)
Data
Data Table 1: Amplitude and Period for L = 1.50 m, m = 50 g
Angle (°)
Time 1 (s)
Time 2 (s)
Time 3 (s)
5
10
20
25.7
25.5
25.2
24.8
25.2
25.1
24.7
24.9
25.3
Average
time (s)
25.1
25.2
25.2
Period (s)
Observations
2.51
2.52
2.52
Fast
Slow
Slower
Data Table 2: Length and Period for θ = 10°, m = 50 g
Length (m)
Time 1 (s)
Time 2 (s)
Time 3 (s)
1.50
1.00
0.50
25.5
20.1
14.7
25.2
19.9
14.7
24.9
20.1
14.8
Average
time (s)
25.2
20.0
14.7
Period (s)
Observations
2.52
2.00
1.47
Slow
Low times
Lowest
Data Table 3: Mass and Period for θ = 10°, L = 1.0 m
Mass (g)
Time 1 (s)
Time 2 (s)
Time 3 (s)
Period (s)
Observations
20.1
Average
time (s)
20
50
20.1
19.9
2.00
20.7
20.6
20.7
2.07
20.9
20.7
20.8
2.08
Average
times
Slightly
slower
Slowest
100
20.6
150
20.9
Data Table 4: Acceleration Due to Gravity for θ = 10°, m = 50 g
Length
(m)
1.50
1.00
0.50
Period
squared
(m/s2)
6.4
4.0
2.2
g
(calculated)
(m/s2)
9.3
9.85
8.9
Δg
% Error
(calculated) (calculated)
(m/s2)
2.8
5.1
2.9
0.5
4.5
9.2
g (graph)
(m/s2)
% Error
(graph)
9.2
12.2
Graph 1: L versus T2
Excel graph, should have L on y-axis, Time2 on x-axis, remember to include units
Data Table 5: Energy for θ = 10°, m = 100 g, L = 1.00 m
Amplitude (m)
0.174
Umax (J)
0.15
Kmax (J)
0.15
% Difference
0
Calculations
4π2 L
= 𝑠𝑙𝑜𝑝𝑒 · 4π2
T2
0.2998 ∗ 4𝜋 2 = 11.0 𝑚/𝑠 2
|11.0 − 9.8|
∗ 100% = 12.2%
9.8
𝐴=𝜃∗𝐿
10 ∗ 𝜋
=(
) ∗ 1𝑚 = 0.174𝑚
180
1 𝑚𝑔
𝑈𝑚𝑎𝑥 = ( ) 𝐴2
2 𝐿
100 ∗ 9.81
= .5 (
) 0.1742 = 0.15𝐽
100
2πA 2
𝐾𝑚𝑎𝑥 = .5𝑚 (
)
T
2π0.174 2
. 5(100) (
) = 0.15𝐽
20
𝑔=
25.7 + 24.8 + 24.7
= 25.1𝑠
3
25.1
= 2.51s
10
𝑚
𝑇 2 = 2.522 = 6.35 ( 2 )
𝑠
4𝜋 2 𝐿
𝑔= 2
𝑇
4𝜋 2 (1.50)
𝑔=
= 9.25 𝑚/𝑠 2
6.4
2𝛥𝑇
𝛥𝐿
𝛥𝑔 = 𝑔 [(
) + ( )]
𝑇
𝐿
2(0.1)
0.4
9.3 [(
)] = 2.8 𝑚/𝑠 2
)+(
6.4
1.50
|9.3 − 9.8|
∗ 100% = 5.1%
9.8
(0.15 − 0.15)
|
| · 100 = 0%
0.15 + 0.15
(
)
2
Summary of error
Within the experiment, there are some sources of error due to human error and possible
miscalculations. Within Data Table 1, 2 and 3, the slowing momentum could affect loss in
time measurements, which can explain slower times or even incomplete swings of the
pendulum. In Data Table 4, the error calculations shown are low enough to confirm that
the gravitational acceleration of the mass is measured correctly to match up to the actual
gravitational pull of 9.81 m/s2. By used of the simulated system in Graph 1, the slope
calculated with component 4π2 gives us the estimated gravitational pull within the given
average length and period2. With all these components in mind, a potential and kinetic
energy Max can be given for each one, both with 0% error due to similar calculation
methods.
In conclusion, the calculations of each data table show little to unavoidable error
due to imperfect setting and incorrect time measurements from lost momentum by air
friction.
Exercise 1
1.
2.
3.
4.
Determine the position in the oscillation where an object in simple harmonic
a. has the greatest speed
i. The mean position is where the velocity is at a maximum.
b. has the greatest acceleration
i. The extreme position is where the acceleration is at its max.
c. experiences the greatest restoring force
i. The extreme position is where the restoring force is at maximum
as well.
d. experiences zero restoring force
i. The mean position is where an object experiences zero restoring
force.
Describe simple harmonic motion, including its cause and appearance. (Make sure
to use your own words and be very specific. And few examples would be helpful.)
a. Simple harmonic motion is an oscillatory motion where restoring force that is
acting on a body is directly proportional to displacement from a mean position.
Use the information listed in Data Tables 1 to 3 to describe how the change in, using
complete sentences:
a. amplitude (angle)
b. length
c. mass affect the period of the pendulum
i. The angle, length, and mass have a varying effect on the period of a
pendulum. The period is independent of mass and amplitude but is
proportional to the square root of the length.
Compare your calculations of g using individual measurements and using the graph
in Graph 1, listed in Data Table 4. Do your measurements including uncertainty fall
within the accepted value? Which method is more accurate? List possible sources of
error in your data and calculations.
5.
6.
a. The most accurate measurement based on the table is when the length of the
pendulum is 1 m. It falls within 0.4% of the accepted value. The possible sources
of error are the inaccurate length of strength, the angle at which the pendulum
is swung, and how the stopwatch is timed.
Was energy conserved during the motion of your pendulum? If not, list some
possible ways energy could have been lost from the pendulum system, making sure
to use complete sentences.
a. Without friction, energy would have been conserved as mechanical energy. With
the friction, the energy is converted into heat or other forms. Energy is lost from
the air and the swing from the pendulum and the attached points.
Why did you measure 10 periods of the pendulum instead of just 1? Would your
measurements be improved by measuring 100 periods instead of 10? Why or why
not? How many periods do you think is the optimal number to measure?
a. I measured 10 periods instead of 1 in order to get more consistent results. With
just one period, the pendulum is exerting max energy, and not being as slowed
down from friction as it would be from 10 periods. My measurements from 100
periods would not be as accurate, as the amount of energy needed to reach 100
periods would not be possible from a 5 or ten degree drop. I think the optimal
number to measure is 10-15.
Extension Question
1.
How would the motion of a pendulum change at high altitude like a high
mountain top? How would the motion change under weightless conditions?
(Make sure to use your own words.)
a. On a mountain top, a reduced gravity will result in longer periods of
oscillation. With weightless conditions, the pendulum would not work
because what keeps the pendulum oscillating is gravity.
Citations
Halliday, D., Resnick, R., & Walker, J. (2018). VitalSource bookshelf online. In
online.vitalsource.com (11th ed.). WileyPlus.
https://online.vitalsource.com/#/books/9781119306856/cfi/6/44
```
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