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Lesson 3.2
Lesson 3.1 |
Lesson 3.2 Simple Harmonic Motion Overview This lesson deals with oscillations and vibrations. On completion of this lesson you should be able to describe simple harmonic motion. You should also be able to explain various terms associated with waves or cycles. You should also be able to explain projected circular motion and why certain waves forms are called sine waves. MINI LAB
CHOICE OF ACTIVITIES
Oscillations and Vibrations Oscillations are vibrations and vibrations are often called oscillations. Oscillations refer primarily to pendulum-like motions. Vibrations are movements backwards and forwards. In order to draw a distinction between oscillations and vibrations, it may be useful to define oscillations as movements that are similar in some respects to the movements of a pendulum. An oscillating object or disturbance moves between two extremes in such a way that the speed of movement reduces as it approaches the extreme. The speed is also at a maximum at the mid-point between the extremes. Vibrators, such as electric bells and buzzers, have a part that moves between two extremes. In such items, the speed of the moving object is at a maximum just before it is abruptly stopped at each of its path. In an electromechanical buzzer, a metal object moves between one position that it is drawn to by a spring and a second position that it is drawn to by an electromagnet. When on, the electromagnet exerts a stronger attractive force than the spring. As the metal object is drawn towards the electromagnet it accelerates. The current to the magnet is switched off just before the object strikes the surface of the electromagnet. The spring then draws the metal object back to the other end of its path and the electromagnet is switched on again. If the position of the vibrating object with reference to one of the extremes is plotted with time, the plot would be similar to the diagram below. This represents a typical "saw-tooth" pattern. The moving object repeatedly accelerates, is brought to an abrupt stop and accelerates in the opposite direction until it is again abruptly stopped.
Pendulums When a simple pendulum swings backward and forward in a small arc, the movement is an example of simple harmonic motion. The process is called simple because it is a basic movement. More complex harmonic motion can be a combination of two or more simple harmonic motions. For example, basic pendulums can display complex motion. A weight on the end of a rope could also swing in elliptical or circular patterns. Complex pendulums with more than one pivotal point will also produce complex movement patterns.
Harmonic Motion Simple harmonic motion is a periodic motion; that is, it repeats itself at regular intervals. Other examples of objects whose motion has the characteristics of simple harmonic motion are a mass that is oscillating up and down at the end of a stretched spring and air molecules vibrating back and forth as a sound wave passes. Harmonic motion occurs when an object with inertia moves under the influence of opposing forces. that work together to produce a regular movement to and fro. Due to the inertia of the object and resulting delays in conversion of energy, the opposing forces cause the object to move back and forth in a repeating pattern. Movement of the object represents repeated conversion of potential energy to kinetic energy and back to potential energy. Pendulums A pendulum moves between two extreme positions that represent points at which it has the maximum amount of potential energy. Gravity causes the pendulum to accelerate from each of these extreme positions towards its normal position at rest but on reaching the lowest point in its swing, the pendulum has gained kinetic energy that carries it past the lowest point. After passing the lowest point, it loses kinetic energy and gains potential energy until it again reaches the extreme position with maximum potential energy and no kinetic energy. Momentum and gravity work together in keeping the pendulum moving until friction has absorbed all of the available energy. If the displacement of a pendulum from its normal position at rest is plotted over time, the graph would be similar to the diagram below.
Projected Circular Motion Simple harmonic motion is the projection of uniform circular motion onto one axis. For example, when a cyclist is moving along an at constant speed while pushing on the pedals of the bicycle, the pedals move in a circular pattern while her knees move up and down in a slight arc that is almost linear. The linear motion of her knees is linked to the circular motion of her feet at the pedals. A similar example of linear motion projected from circular motion is the movement of a piston in an engine as the crankshaft rotates. In both of these cases the projection is slightly modified because the distance between the point on the circle and the projected point onto the axis (line) is changed by the changing angle of the connecting link. A more accurate example of a projection of circular motion would be the movement of an object that is connected via a slot to pin on a rotating wheel. In the diagram below, the horizontal bar moves back and forth horizontally as the wheel rotates. The pin on the wheel moves up and down in the slot connected to the horizontal bar.
If an object is linked to a rotating point that follows a circular path with a diameter of 0.1 meters, the distance between the two extremes of the object’s movement will be 0.1 meters. As the link rotates, the position of the object between these two extremes will be related to the sine of the angle of rotation. The Sine Wave The variation of the sine of the angle of rotation with time for a rotating body can be plotted as follows:
If a simple wave is moving through a solid, liquid or gaseous medium, the displacement of particles of the medium along the path of the wave could be represented by the first diagram below. The second diagram shows the displacement of an individual particle with time as the wave passes.
Each of the diagrams above is similar in shape to the plot of the sine of the angle of rotation with time for a rotating body. Because of this similarity, simple waves are often called sine waves and the pattern or movement that repeats is called a cycle. A cycle can be represented by the minimum distance or time between two identical points in the diagram. For points to be identical, they must also be related to the same direction of movement. For example, Points A and A’ are identical, as are points B and B’. They represent points in the same phase of the cycle. Points A and B on the diagram below are not identical. If related to a swinging pendulum, the pendulum would be swinging in opposite directions at these two points. The points are out of phase by 180º of rotation, 50% of a cycle or 50% of a wavelength – depending on the variable being plotted. Wavelength Identical points on a wave diagram represent points that are in phase. If the displacement of particles of the medium along the path of the wave is plotted with distance, the wavelength is the distance between identical points on adjacent cycles.
Amplitude The amplitude of a wave or oscillation is the maximum value of the disturbance in the medium. For particles oscillating in a medium as waves pass, the amplitude is the maximum distance moved from the normal position at rest. Since oscillating particles can move in opposite directions from their normal position at rest, a sign convention is use to differentiate between movements in the two directions. For example, the distance of a pendulum to the right of it’s normal position at rest could be given a positive value whereas the distance to the left could be assigned a negative value. The nature of the sign convention needs to be defined when plotting graphs or performing calculations involving simple harmonic motion. The diagram below shows the variation of the amplitude (plotted on the y-axis) of a particle in a medium with time (along the x-axis) as the particle oscillates under the influence of waves moving through the medium.
Simple Harmonic Motion Review Questions
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