Background   In the world of nature we often find processes and events which periodically repeat themselves in time. After a certain time interval, or period, the system or process returns to its initial condition. The tides in the ocean, the positions of the sun, moon and planets and lunar eclipses, and a host of other astronomical phenomena recur with a more or less complicated regularity. The study of simpler systems allows us to obtain information concerning many of the others. One such simple system is the pendulum.   A typical simple pendulum consists of a heavy pendulum bob (mass = M) suspended from a light string. It is generally assumed that the mass of the string is negligible. If the bob is pulled away from the vertical with some angle, q, and released so that the pendulum swings within a vertical plane, the period of the pendulum is given as:   T = 2p (L/g)1/2(1 + 1/4 sin2q/2 + 9/16sin4q/2 + ...)            Equation 1   where L is the length of the pendulum and g is the acceleration due to earth's gravity. Note only the first three terms in the infinite series is

given in the above equation. The period is defined as the time required for the pendulum to complete one oscillation. That is, if the pendulum is released at some point, P, the period is defined as the time required for the pendulum to swing along its path and return to point P.

The above formula for the pendulum's period is greatly simplified if we limit the initial angle q to small values. If q is small, we can approximate the period of the pendulum with the following expression.

      T = 2p (L/g)^1/2         Equation 2

Note that the period in this expression is independent of the pendulum's mass as initial angle, q. It is important to understand that the above equation is valid only in the small angle approximation. If the magnitude of the displacement is sufficiently small, and if we neglet the effects of friction and air resistance, the motion of the mass is relatively simple and is called simple harmonic motion.

If we allow the mass to move in one plane only we have what is called a simple pendulum, a device famous for its use in the grandfather clock. The mass m is called the bob of the pendulum. At time t = 0, we displace the mass a distance R to the right, as measured from the vertical or equilibrium position, and then release it. Consider the forces acting upon m just after it is released. There is the weight, w = mg, pulling downward and the tension T in the string as shown on the right. It is seen that there is a net force acting to move the mass back to its original position. The restring force F is proportional to the displacement, so long as the displacement is small, and it always points in the direction of the equilibrium position. Since there is an unbalanced force acting on m, Newton's second law tells us that the object will be accelerate in the direction of F. The bob thus returns to the equilibrium position and, when it arrives there, it has a velocity. Inertia then carries the bob beyond the equilibrium position. The process repeats and the bob is returned to its original position (ignoring friction, etc). The time required for one complete cycle is called the period.  The number of cycles per unit time is the frequency.

Setup

Equipment Scale For weighing things Cord For suspending things Protractor For measuring angles Bobs To serve as the bobs Level To level things Timer To time things Ruler To measure things

Objectives

1.  Determine the maximum angle for which the equation   T = 2p (L/g)^1/2  is valid. In other words, find the cutoff angle for which the small angle approximation fails.

     Obtain results for angles ranging from 5 to 45 degrees.

2.  Use the simple pendulum to determine the value of g, the acceleration due to Earth's gravity.

3.  Explain why the motion of the pendulum is harmonic motion but not simple harmonic motion.

4.  Show that equation 2 above is dimensionally correct (use dimensional analysis).

5.  Use 5 appropriate data points to answer the following questions. For each case, identify the variables(s) you held constant.

      What is the relationship between:

     a.  mass of the bob and the period

     b.  length of the cord and the period

     c.  initial displacement of the bob and the period 

6.  Assume that the bob is raised to a point 20 degrees from the equilibrium position.

     a.  What is the work required to accomplish this?

     b.  What is the potential energy at this point?

     c.   Assume that the bob is released released from the above point and allowed to swing. Neglecting friction and air resistance, what is the velocity of the bob when it reaches

           the equilibrium position? 

    d.  Assume that the cord is released just as the bob reaches the equilibrium position. Describe the trajectory of the bob immediately after release.