PRELAB
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PURPOSE
To investigate the characteristics of simple harmonic motion by analyzing the vertical oscillations of a mass suspended from a spring.
EQUIPMENT spring, spring holder, assorted masses, mass holder, Vernier motion detector
RELEVANT EQUATIONS
The motion of simple harmonic motion can be simply described in terms of the basic trigonometric functions like sine and cosine. These relations are summarized below:
Term | Symbol | Units | ||
Period | T | sec | ||
Frequency | f = 1/T | Hz | ||
Angular frequency | ω = 2πf | sec^{-1} | ||
Spring constant | k = F/x | N/m | ||
Kinematic Equations | ||||
Displacement | x = A cos ωt | meters | ||
Velocity | v = -ωA sin ωt | m/sec | ||
Acceleration | a = -ω^{2}A cos ωt | m/sec^{2} | ||
Dynamic Relations | ||||
Force | F = - k x | N | ||
Angular freq | ω = | sec^{-1} | ||
Period | T = 2π | sec | ||
Spring constant | k = m ω^{2} | N/m |
KE_{max} = mv^{2}_{max}
= m ω^{2}
A^{2} = k
A^{2} = PE_{max}
DISCUSSION
An object that exhibits periodic motion will repeat its motion within a certain time interval. Periodic motion may or may not be sinusoidal. If it is, we call it simple harmonic motion (SHM). SHM has one special property: the displacement from equilibrium and the acceleration are directly proportional but oppositely directed. We will want to look for this property and for any other interesting connections we can find.
Simple harmonic motion occurs in many places in the natural world. The specific SHM that you will study in this experiment is that of the mass-on-a-spring oscillator. When a mass is suspended from a vertical spring, the spring is stretched. The ratio of the weight applied mg, to the amount of elongation Δx, is called the spring constant of the particular spring. A mass suspended from a spring at rest is in equilibrium-the net force on it is zero, because the spring supplies a force that is equal but opposite in direction to its weight. The spring force is directly proportional to its elongation (or its compression), and so we can write:
The minus sign tells us that the spring force is in the direction opposite to the Δx. If the spring is elongated, Δx is downward, then the spring force is directed upward. If the spring is compressed Δx is upward, then the spring force is directed downward. This opposing tendency is called a restoring force, because the spring always tries to restore the mass back to its equilibrium position.
Suppose the mass is displaced a bit below its equilibrium position by some external agent. This act requires an amount of external force above and beyond the weight of the mass, because with the additional elongation, the spring is exerting a greater force upward.
If the mass is then released by the external agent, there is an unbalanced force: the restoring force of the spring is greater than the weight of the mass, and so the mass is accelerated. As it accelerates, it picks up speed and moves towards the equilibrium point. By the time it gets there, it is moving rather fast and of course at that exact point, the net force becomes zero because that is where the spring force just exactly balances the weight of the mass. So, at that instant, there is zero acceleration, but the mass has some kinetic energy built up so that it continues to move in the upward direction and begins to compress the spring. When this happens, the spring force acts downward in the direction opposite to the velocity, and so the mass suffers a deceleration (negative acceleration), i.e. it slows down.
As it continues to compress the spring the restoring force gets larger and larger, until finally the point is reached where the mass reaches zero velocity; it is instantaneously at rest. Here however, the force is unbalanced again and the net force is directed downward, so the mass starts moving in that direction, back through the equilibrium point and back to the downward limit where it started its motion.
As you can see, the initial conditions are the same, so the mass will repeat its motion as before. This type of repetitive motion is called an oscillation. The time for one complete oscillation is called the period. The rate of the number of oscillations per unit time is the reciprocal of the period and is called the frequency.
The maximum elongation or compression of the spring (they are equal) is called the amplitude of the oscillation. In simple harmonic motion, the period and frequency are completely independent of the amplitude. In other words, no matter how far the mass is initially displaced, the period and thus the frequency of the motion stays the same and is related only to the physical properties of the system, such as the spring constant and the mass. The period, T, for the mass-on-a-spring oscillator is given by
(1) |
Its reciprocal, the frequency, is therefore expressed as:
(2) |
Figure 19-1: Schematic Diagram of the Apparatus
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