Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called simple harmonic motion, or SHM. The position of the oscillating object varies sinusoidally with time. Many objects oscillate back and forth. The motion of a child on a swing can be approximated to be sinusoidal and can therefore be considered as simple harmonic motion. Some complicated motions like turbulent water waves are not considered simple harmonic motion. When an object is in simple harmonic motion, the rate at which it oscillates back and forth as well as its position with respect to time can be easily determined. In this lab, you will analyze a simple pendulum and a spring-mass system, both of which exhibit simple harmonic motion. A particle that vibrates vertically in simple harmonic motion moves up and down between two extremes y = ±A. The maximum displacement A is called the amplitude. This motion is shown graphically in the position-versus-time plot in Figure 1. One complete oscillation or cycle or vibration is the motion from, for example, to and back again to The time interval T required to complete one oscillation is called the period. A related quantity is the frequency f, which is the number of vibrations the system makes per unit of time. The frequency is the reciprocal of the period and is measured in units of Hertz, abbreviated Hz; If a particle is oscillating along the y-axis, its location on the y-axis at any given instant of time t, measured from the start of the oscillation is given by the equation Recall that the velocity of the object is the first derivative and the acceleration the second derivative of the displacement function with respect to time. The velocity v and the acceleration a of the particle at time t are given by the following.
( 4 ) Notice that the velocity and acceleration are also sinusoidal. However, the velocity function has a 90° or π/2 phase difference while the acceleration function has a 180° or π phase difference relative to the displacement function. For example, when the displacement is positive maximum, the velocity is zero and the acceleration is negative maximum. Substituting from equation 2 into equation 4 yields From equation 5, we see that the acceleration of an object in SHM is proportional to the displacement and opposite in sign. This is a basic property of any object undergoing simple harmonic motion. Consider several critical points in a cycle as in the case of a spring-mass system in oscillation. A spring-mass system consists of a mass attached to the end of a spring that is suspended from a stand. The mass is pulled down by a small amount and released to make the spring and mass oscillate in the vertical plane. Figure 2 shows five critical points as the mass on a spring goes through a complete cycle. The equilibrium position for a spring-mass system is the position of the mass when the spring is neither stretched nor compressed. The mass completes an entire cycle as it goes from position A to position E. A description of each position is as follows.
Position A: The spring is compressed; the mass is above the equilibrium point at and is about to be released. Position B: The mass is in downward motion as it passes through the equilibrium point. Position C: The mass is momentarily at rest at the lowest point before starting on its upward motion. Position D: The mass is in upward motion as it passes through the equilibrium point. Position E: The mass is momentarily at rest at the highest point before starting back down again.
( 10 ) f =
( 11 ) T = 2π
( 13 ) where the negative sign implies that the restoring force acts opposite to the direction of motion of the bob. Since the bob is moving along the arc of a circle, the angular acceleration is given by From equation 13 we get In Figure 9, the blue solid line is a plot of sin(θ) versus θ, and the straight line is a plot of θ in degrees versus θ in radians. For small angles, these two curves are almost indistinguishable. Therefore, as long as the displacement θ is small, we can use the approximation sin θ ≈ θ. With this approximation, equation 14 becomes Equation 15 shows the (angular) acceleration to be proportional to the negative of the (angular) displacement, and therefore the motion of the bob is simple harmonic and we can apply equation 5 to get Combining equation 15 and equation 16 and simplifying, we get
( 17 ) f =
( 18 ) T = 2π |