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