Download - HSC Physics Practical 1
HSC Physics Practical 1
Aim: To investigate the oscillations of a simple pendulum
Theory: Simple harmonic motion (SHM) is a periodic motion that is neither driven nor
damped. A force is experienced by an object in simple harmonic motion, which is
directly proportional to and in the opposite direction with its displacement from the
equilibrium position. From that, we can conclude that it obeys Hooke’s Law. From
Newton’s Second Law of motion, F= ddt
(mv) and since the motion obeys Hooke’s
Law, thus F=-kx, where F is the restoring elastic force experienced by the spring in the
system, k is the spring constant (N m-1), and x is the displacement from the equilibrium
position (in m).
As the pendulum swings, it is accelerating both centripetally, towards the point of
suspension and tangentially, towards its equilibrium position. It is its linear, tangential
acceleration that connects a pendulum with simple harmonic motion.
The weight component, mg sinθ, is accelerating the mass towards equilibrium along
the arc of the circle. This component is called the restoring force, F restoring of the
pendulum.
Since the arc length, s, is directly proportional to the magnitude of the central angle, θ,
according to the formula s = rθ, Therefore, the radius of the circle, r, is equal to L, the
length of the pendulum. Thus, s = lθ, where θ must be measured in radians.
Substituting into the equation for SHM, we get F restoring= - ks and
mg sinθ = - k(lθ)
Solving for the "spring constant" or k for a pendulum yields
mg sinθ = k(lθ)
k = mg sinθ
lθ
Since the value of sin x approximates the value of x for small angles; that is,
limx →0 ( sinx
x )=1❑
Or, equivalently, for x equal to small values, the value of sin x would approach that of x.
sin x = x - x3
3! + x5
5! - x7
7 ! + …..
Using this relationship allows us to reduce our expression for the pendulum's "spring
constant" to
k ≈mgθlθ
∴k ≈ mgl
Substituting this value for k into the SHM equation for the period of an oscillating system
to obtain
T = 2π √ mk
T = 2π √ mmgl
T = 2π √ lg
Since 1t= p
√ l− p
q , compare this equation with the general equation Y= mX + C, thus
Y = 1t
X=1
√l m=p C= -
pq
∴ the value of p is equal to m, which is the gradient of the graph of 1t
against 1
√l.
q = - pC
= - p
value of y−intercept
Apparatus: Split rubber stoppers, thread, pendulum bobs, stopwatch, retort stands, ruler.
Procedure:
In this experiment, the motion of two simple pendulums, and the interval between successive
times at which the pendulums are moving together. How this time interval is affected when the
length of one of the pendulums is changed is investigated.
1. Two simple pendulums are set up side by side as shown in Figure 1.1, with each string
clamped between two split rubber stoppers. The length of pendulum A is set to about 0.65m.
Pendulum A is left at its set length throughout the experiment.
2. Pendulum B is adjusted so that its length, l is about 0.5m. The value of l is measured and
recorded.
3. Both pendulums are set into motion with small oscillations. The stopwatch is started when
the two pendulums are lined up as shown in Figure 1.2 and are moving in the same direction.
4. The time, t that elapses before the next occasion when the two pendulums are lined up and
moving in the same direction are determined.
5. l is changed and step 4 is repeated until six sets of values for l and t is obtained. l is set from
about 0.3m to about 0.6m.
6.1t= p
√ l− p
q is the equation relating t and l where p and q are constants.
Tabulation of Data:
Length of
pendulum(m)
1st
reading(s)
2nd
reading(s)
3rd
reading(s)
Average
reading(s)
1t
(s−1 ¿1
√l(m−1 ¿
0.30 3.62 3.50 3.12 3.413 0.293 1.826
0.35 4.37 4.44 4.25 4.353 0.230 1.690
0.40 5.81 5.56 5.75 5.707 0.175 1.581
0.45 7.75 7.62 7.69 7.687 0.130 1.491
0.50 13.28 12.00 11.00 12.093 0.083 1.414
0.60 37.25 37.12 37.50 37.290 0.027 1.291
Graph:
1. A graph of 1t
on the y-axis against 1
√l on the x-axis is plotted and the line of best fit is drawn.
2. The gradient and y-intercept of this line is determined.
From the graph, the gradient = 0.275−0.027
1.79−1.28 = 0.4863 s−1 m
12
A point (1.690, 0.230) is substituted into the equation, 1t= p
√ l− p
q ,
∴ 1.690 = 0.4863 (0.230) - C
C = -1.578 s−1
Y-intercept = -1.578 s−1
Calculation:
Since m, gradient = 0.4863 s−1 m12 , and m = p, then p = 0.4863
C= -pq
q = - pC
, so q = - 0.4863−1.578
= - 0.3082
Conclusion:
Value of p = 0.4863
Value of q = -0.3082
Discussion:
1. When this experiment is conducted, some problems have been encountered. During the
experiment, the readings are recorded by a person when the two pendulums are lined up and
moving in the same direction. By using this method, random error may occur as the person’s
reaction time may be different. Besides that, when the pendulum bobs are released, the
motion of the pendulum bobs may not be parallel. This reduces the accuracy of the results.
2. The three possible sources of errors are the air flow ( wind ) may affect the accuracy of the
readings as it will produce excessive unwanted force on the pendulum bobs. Besides that, the
angle from which the pendulum bobs are released, may not be the same. This will affect the
time taken for the two pendulums to line up and moving in the same direction.
3. To improve my results, certain precaution have to be taken, that is ensure that both the
pendulum bobs do not collide with each other as well as the apparatus during the experiment.
Besides that, both pendulums are set into motion with small oscillations, that is at an angle
smaller than 10o. Other than that, the stopwatch reading is taken immediately when the two
pendulums are lined up and moving in the same direction. The length of the pendulum bob
may not be measured correctly, that is from the center of the pendulum bob to the rubber
stoppers. This is because the center of the pendulum cannot be determined accurately.
References:
Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems
(5th ed.). Brooks Cole. ISBN 0-534-40896-6.
John R Taylor (2005). Classical Mechanics. University Science Books. ISBN 1-891389-22-
X.
Grant R. Fowles, George L. Cassiday (2005). Analytical mechanics (7th ed.). Thomson
Brooks/Cole. ISBN 0-534-49492-7.
Walker, Jearl (2011). Principles of physics (9th ed.). Hoboken, N.J. : Wiley. ISBN 0-470-
56158-0.
(n.d.). Retrieved from • http://en.wikipedia.org/wiki/Simple_harmonic_motion
Catharine, H.Colwell. (n.d.). Derivation: period of a simple pendulum. Retrieved from
http://dev.physicslab.org/Document.aspx?
doctype=3&filename=OscillatoryMotion_PendulumSHM.xml