Hooke’s Law and Simple Harmonic Motion
Name: Ryan McKiernan
Lab partner: Han Nguyen
TA: Gejian Zhao
Tuesday 10:25-12:15
Abstract
The experiment was done to measure spring constants, test simple harmonic
motion, and verify the conservation of energy lab. Springs were tested with Hooke’s
Law using a mass hanger with parallel springs and a series. Also, simple harmonic
motion was performed with a collision car and two springs. Major numerical findings
were 31.88 ± 0.096 and 33.96 ± 0.096 spring constants compared to the theoretical
value of 32. The parallel springs had a k value of 64.7 ± 0.14 and the series of springs
have a k value half of the theoretical, at 16.6 ± 0.040. The spring constant for the
oscillating car was 68.03 ± 0.128. The spring constants agreed with theoretical
predictions because percent discrepancies were low.
Objective:
The objective of the experiment was to measure the spring constant of the
springs using Hooke’s Law, explore springs with harmonics, and to verify the
conservation of energy law.
Equipment:
The equipment used in the experiment was two identical springs, a long rubber
band, support stand with a meter stick, 50-g mass hanger, a set of masses from 100-g
up to 600-g, a balance scale, and the science workshop interface with force sensor and
rotational motion detector used as a linear sensor. For part two, a collision cart, motion
sensor, rubber bands, and two rectangular weights were needed.
Procedure:
The experiment was done in two parts. In part one, a spring was hung from the
support and a weight hanger was hung. Masses were added 100g at a time and the
position was measured for each mass. In the second part of part one, the rotational
motion sensor was used with a string to pull on the spring until 10N was shown on the
computer. This was done with two parallel springs as well as two conjoined springs.
Next, a collision cart was placed on a level track with a spring on each side stretched
and was pushed gently to show harmonic motion. Different increments of force were
used for each trial.
Data Analysis
Part A: One Spring
1. kexp,1 = 33.96 ± 1.285
2.
(𝒆𝒙𝒑−𝒕𝒉𝒆𝒐𝒓)
𝒕𝒉𝒆𝒐𝒓
× 𝟏𝟎𝟎% =
(𝟑𝟑.𝟗𝟔 −𝟑𝟐)
𝟑𝟐
kP.D,1 = 6.13%
3.
1.285
33.96
× 100% = 3.78%
kR.E.,1 = 3.78%
4. kexp,2 = 31.88 ± 0.096
5.
(31.88−32)
32
× 100% = 0.375%
KP.D.,2 = 0.375%
6.
0.096
31.88
× 100% = 0.301%
kR.E.,2 = 0.301%
Part B: Two Springs
1. kp = 64.7 ± 0.14 (2k)
2. ks = 16.6 ± 0.040 (1/2k)
× 𝟏𝟎𝟎% = 𝟔. 𝟏𝟑%
Part C: Oscillating Cart
1. 𝑘 = 𝜔2 𝑚
𝑘=(
2𝜋 2
) 𝑚
𝑇
2
2𝜋
𝑁
2. 𝑘 = (0.566𝑠) (0.5039𝑘𝑔) = 62.1 𝑚
3. TA = 0.123 m
1
PEmax = 2 𝑘𝐴2
1
𝑁
PEmax = 2 (62.1 𝑚)(0.123𝑚)2 = 0.469 𝐽
1
4. KEmax = 2 𝑚𝑣 2 Tv = 0.751 m/s
1
𝐾𝐸𝑚𝑎𝑥 =
5.
2
(0.422𝐽−0.469𝐽)
0.469𝐽
𝑚
(1.504𝑘𝑔)(0.751 𝑠 )2 = 0.422 𝐽
× 100% = 10.02%
ER.L. = 10.02%
2𝜋
2
6. 𝑘 = (𝑠𝑙𝑜𝑝𝑒)
𝑘=(
2𝜋 2
) = 68.03 𝑁/𝑚
0.7618
∆𝑠𝑙𝑜𝑝𝑒
7. ∆𝑘 = 2𝑘( 𝑠𝑙𝑜𝑝𝑒 )
𝑁
7.15𝑒−5
8. ∆𝑘 = 2(68.03 𝑀)( 0.7618 ) = 0.12
Results
Part A
Spring
1
Spring
2
Experimental spring
constant k
(N/m)
Theoretical spring
constant k
(N/m)
Percent
Difference (%)
Relative Error (%)
33.96 ± 1.285
32.00
6.13%
3.78%
31.88 ± 0.096
32.00
0.375%
0.301%
Part B
Experimental spring constant k
(N/m)
Spring in series
16.6 ± 0.040
Spring in parallel
64.7 ± 0.14
Part C
Experimental spring constant k
(N/m)
Oscillating Cart
68.03 ± 0.128
Discussion
Hooke’s Law states that if a stretch of a spring is small, the magnitude of the
elastic force is directly proportional to the stretch. F = -kx. Simple harmonic motion
states that if a hanging mass is displaced from the equilibrium position and released,
simple harmonic motion will occur. The position changes with a sinusoidal dependence
on time. Harmonic motion contains a k value when dealing with springs, as in the
experiment.
Based on the graphs for F vs. Δx, the springs do obey Hooke’s law. The
theoretical value for k was 32 N/m and both springs were close to that with values of
33.96 ± 1.285 and 31.88 ± 0.096 with percent discrepancies of 6.13% and 0.375%.
These discrepancies were quite low which means the experiment obeyed Hooke’s Law.
For a system of two springs, the k values directly relate to the individual spring
constants because the parallel springs have a k value that is around double, 64.7 ± 0.14
and the series of springs have a k value half of the theoretical, at 16.6 ± 0.040. This
makes sense because two springs were used to produce k values that were 1/2k and
2k. With the rubber band, the graph was linear implying the k constant was involved and
could be used for Hooke’s Law. A conclusion that can be made about the relationship
between period and amplitude based on the collected data is that amplitude does not
affect period. The amplitude increased, but the period stayed the same. The Law of
Conservation of Energy was upheld in the experiment because potential energy was
close in value to kinetic energy. The values were 0.469J and 0.422J respectively.
10.02% was lost, which is relatively low. Overall, the main characteristics of simple
harmonic motion were shown because the sinusoidal graphs for velocity and position
versus time were created and proved that it relied on time. A mass was displaced and
released, and simple harmonic motion occurred.
Conclusion
The objective of the lab was met because the spring constant was measured in
many different scenarios with Hooke’s Law, simple harmonic motion was performed,
and the Law of Conservation of Energy was verified. The spring constant remains the
same for a single spring regardless of the changes made to it. Also, simple harmonic
motion correlates with Hooke’s Law.