Time interval between successive dots = 3(1/50) =0.06s
- The data for time and displacement were entered in columns A and B respectively in a spreadsheet program.
The following formulae were entered in the cells flagged:
The cells were copied to the other cells in the same volume.
By approximation, the average velocity and average acceleration were taken as the instantaneous velocity and instantaneous acceleration.
-
Using the spreadsheet program, the displacement-time (x-t), velocity-time (v-t) and displacement-acceleration(x-a) graphs were plotted.
Theories:
A simple harmonic motion (SHM) is an oscillation with the following properties:
- The period (or frequency) of oscillation is independent of the amplitude of the motion.
When a pendulum is lifted by an angle θ from the vertical, it experiences a total restoring force of –mg sin θ .
When θ < 10o, if θ is measured in radians.
Then
Therefore, the period of the oscillation is
which depends on the length of string and gravitational field strength only.
-
The acceleration a is directly proportional to the displacement x and is always directed towards the central point of the oscillation. If the displacement is positive, then the acceleration is negative, and vice versa. Mathematically this can be expressed as
a = -kx
where k is a constant.
The purpose of this experiment is to study the motion of a simple pendulum—to find out how its displacement x, velocity v and acceleration a vary with time. The simple pendulum is a simple harmonic oscillator if the relationship a = -kx is shown.
Measurements:
-
Displacement-time (x-t) graph:
-
Velocity-time (v-t) graph:
-
Displacement-acceleration(x-a) graph:
Discussion:
-
For the x-t graph, half of a cosine curve is obtained. From it, the amplitude and the period of the simple harmonic motion can be found.
-The amplitude A = 0.37m
-The period T = 2(1.02) = 2.04 s
For the v-t graph, half of a negative sine curve is obtained. From it, the maximum velocity of the simple harmonic motion can be found.
-The maximum velocity vmax = 0.98333ms-1
For the a-x graph, a straight line with negative slope passes through zero is obtained. From it, it was found that the acceleration of pendulum was directly proportional to its displacement from a fixed point and is always directed towards that point, hence the motion is simple harmonic.
-The experimental value of T = 2(1.02) = 2.04s
-The theoretical value of T =2π√ (1.5/10) = 2.43 s
% error in T = (2.43-2.03)/2.43 x 100% = 16.5%
- Measure from the bottom of the heavy stuff to the centre of the ringed mass.
- Do not cause the pendulum to swing through an angle greater than 10 degrees.
- Ensure that the pendulum swings in one plane only, so as to avoid circular movements.
Conclusion:
Simple harmonic motion is a periodic motion which the period of oscillation is independent of the amplitude of the motion. Also, the acceleration of the body is directly proportional to its displacement from a fixed point and is always directed towards that point in a simple harmonic motion. Therefore the equation relating acceleration and displacement can be written as a= - (constant) x.
