Physic lab report - study the simple harmonic motion (SHM) of a simple pendulum and to investigate the phase relationship between the displacement, velocity and acceleration, and to investigate how acceleration is related to displacement in a simple harmo

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  Ivan Liu Chun Pok

6A(11)

Group 8

Laboratory report: Studying a simple harmonic oscillation

Objectives:

To study the simple harmonic motion (SHM) of a simple pendulum and to investigate the phase relationship between the displacement, velocity and acceleration, and to investigate how acceleration is related to displacement in a simple harmonic motion.

Apparatus:

  • half metre rule
  • a light string
  • pendulum bob
  • video camera with tripod stand
  • computer with Motion Video Analysis (MVA) software and Microsoft Excel installed

Experimental design:

Fig. 0        

        

Theory:

For an object or mass moving in a simple harmonic motion, the displacement, velocity and acceleration change periodically in both magnitude and direction. The acceleration in particular is always proportional to its displacement from the equilibrium position and must always be directed towards the equilibrium point. Mathematically it can be expressed as
a = -kx,         where k is a constant and x is the displacement from the equilibrium point.

Also for a simple harmonic oscillation, the period or frequency of oscillation is independent of the amplitude of the motion.

In Figure 1, x is the displacement of the pendulum bob from the equilibrium point Q. Points P and R are points where the maximum displacement (amplitude A) can be obtained. Theoretically, the following equations are true for S.H.M.:
When the motion starts at the equilibrium position (point Q)

x = A sin ωt                where ω is angular velocity
v
= ω A cos ωt
a
= - ω2A sin ωt

Period T = 2 π / ω                        

                Fig. 1

When the motion starts at the position
where the amplitude is obtained (point P or R)

x = A cos ωt                where ω is angular velocity
v
= ω A sin ωt
a
= - ω2A cos ωt

Period T = 2 π / ω                                                

In theory, the displacement-time, velocity-time and acceleration-time graphs should be in a sine or cosine curve. Moreover, the velocity graph should lead the displacement by a quarter of the cycle (θ = 90°), and the acceleration graph should lead the velocity by also a quarter of the cycle. This can be illustrated by the fig. 2(a), (b) and (c).

Fig. 2(a): Graph of displacement x against time (Suppose the motion starts at the point where lower amplitude is obtained)

Fig. 2(b): Graph of velocity v against time


Fig.2(c): Graph of acceleration a against time

Procedure:

  1. The set-up was assembled in the following procedures:
  1. One end of the spring is clamped firmly on the stand.
  2. The ringed mass was attached to the other end of the spring.
  3. A half-metre rule was clamped on the stand beside the spring and mass such that the top of the half-metre rule corresponds to the top of the spring. (Refer to Fig. 0)
  4. The equilibrium position was marked by a sticker.
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  1. Take readings by using the apparatus in the following procedures:
  1. Student holding the white foam board (Student A)
  1. Hold the white foam board behind the set up so that the movement of the spring system is not disturbed by any other backgrounds.

  1. Student conducting the experiment (Student B)
  1. Stand beside the set up. Make sure that the spring system at equilibrium is in a steady and stable condition.
  2. When the video taking was on, pull down the spring some distance (e.g. about 5 cm) and set the spring moving.
  3. Make sure the ...

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