Simple Harmonic Motion

SIMPLE HARMONIC MOTION: PENDULUM SCIENTIFIC CONCEPTS Simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law, that is, the force is directly proportional to the displacement x and points in the opposite direction. The motion is periodic, the body oscillates about an equilibrium position in a sinusoidal pattern. Each oscillation is identical, and thus the period, frequency, and amplitude of the motion are constant. If the equilibrium position is taken to be zero, the displacement x of the body at any time t is given by where A is the amplitude, f is the frequency, and φ is the phase. The frequency of the motion is determined by the intrinsic properties of the system often the mass of the body and a force constant, while the amplitude and phase are determined by the initial conditions like displacement and velocity of the system. The kinetic and potential energies of the system are also determined by these properties and conditions. One of the phenomena that can be approximated by simple harmonic motion is the motion of a pendulum. A pendulum is any mass which swings back and forth on a rope, string, or chain. Pendulums can be found in old clocks and other machinery. A playground swing is a pendulum. If you pull the mass away from its rest position, so that the string is at an angle, and then let go, the mass will begin to swing back and forth. The length of time it takes the mass to swing all the way over and back, once is called the period of the pendulum. In these experiments, the dependent variable will always be the time for one full swing, or the period. The three independent variables will be the mass, the angle, and the length of string. The controlled variables will be the attachment point of the string, the string itself, the method used to time the pendulum, and the variables we are not currently testing. These will remain the same for each test, so that we know they won't affect the results. In this experiment, the variable that we will test is the length of string. Experiment result: length of string 20 30 40 45 50 55

trial 1 32.72 31.56 30.58 30.26 29.24 27.82

trial 2 32.99 31.77 30.59 30.12 28.27 27.91

trial 3 32.89 31.66 30.84 30.41 28.84 27.86

average 32.87 31.66 30.67 30.26 28.78 27.86

UNIQUE FEATURE OF THIS ACTIVITY 1. This data can be manipulated easily. For example, we can just enter the data in the table the time of one oscillation; the value of the average can be calculated. 2. Data displayed in a systematic manner, so that we save a lot of time in drawing graph and show relationship between time and the length of string. 3. Spreadsheet is very practical for repetitive calculation. For example, the point of graph will also automatically change according to the change of the data. 4. So that, we do not waste our time recalculation and re-graphing and can continue to do more important things like analyzing the data or graph.

ENGAGE

This is a grandfather clock. What do you think the application used by grandfather clock when every hour, the bell move?

EMPOWER

Steps: 1) Students are given: • • • • • • • • •

One retort stand One G-clamp Two bosses Pendulum bob 100cm thin flexible string Two small blocks of wood Meter rule Half meter rule Stop watch

2) Students are required to plan an experiment to determine the effect of length of strength on the one oscillation. 3) Students have to construct hypothesis in this experiment. 4) You may give this instruction to construct the activity: • • • •

Decide what angle you will use to set the pendulum swinging. Mark it on the wall behind the release point. Set up the full length of string with one weight on the end. Length= 20 cm, 30 cm, 40 cm, 45 cm, 50 cm, 55 cm Make and record measurements to determine the period of these oscillation. Generate your own table.

How to set up the spreadsheet: 1. 2. 3. 4.

Make a columns Enter the data from experiment Copy columns B, C and D Move to cell E and enter the formula = (B+C+D)/(3)

How to draw a graph with your spreadsheet: 1. 2. 3. 4. 5. 6.

Highlights cells E2 to E7 Get the program to draw a graph and choose the graph desired Choose graph lines Name the chart Label the x-axis as length of string and y-axis as time Print out the spreadsheet

Questions: 1) Calculate the value of T using this given formula: T=π g

x

2

T

2π +

y

g

ENHANCE

What do you think of the relationship between the length of pendulum with sound produces?

Answer:

The pendulum swings with a period that varies with the square root of its effective length. The rate of pendulum clocks is adjusted by moving the pendulum bob up or down on its

rod, often by means of an adjusting nut under the bob. To keep time accurately, pendulums are usually made to not vary in length as the temperature changes. Owing to the expansion of metal, the length of a simple pendulum will vary with temperature, slowing the clock as the temperature rises. Another way to keep time accurately is by make sure that pendulum clocks must be at absolutely level. This condition can often be heard audibly in the ticking sound of the clock. The ticks or 'beats' should be at precisely equally spaced intervals, if they are not, and have the sound "tick-tock...tick-tock..." the clock is out of beat and needs to be leveled. While a pendulum does not produce a sound when it oscillates, it does illustrate an important principle. A pendulum consisting of a longer string vibrates with a longer period and thus a lower frequency. Once more, there is an inverse relationship between the length of the vibrating object and the natural frequency at which the object vibrates. Every time the pendulum goes back and forth, it moves a gear one notch. Gears are then used to move the hands of the clock.