A Simple Pendulum
Simple Pendulum
PURPOSE The purpose of this experiment is to study how the period of a pendulum depends on length, mass, and amplitude of the swing.
THEORY A simple pendulum is an idealized model consisting of a point mass (sometimes called a pendulum bob) suspended by a massless unstretchable string. When the bob is pulled to one side of its straight down equilibrium position and released, it oscillates about the equilibrium position. The path of the bob is not a straight line but an arc of a circle with radius L equal to the length of the string. We use as our coordinate the distance x measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to x or to θ because x=L θ. The restoring force F is the tangential component of the gravity. The restoring force is proportional not to θ but sin θ, so the motion is not simple harmonic. However, if the angle θ is small, sin θ is very nearly equal to θ (in radians). Thus we have:
The bob will then execute simple harmonic motion and x (or θ) will be a sinusoidal function of time:
Where A, called the amplitude, is the maximum value of θ (i.e., A=θmax, or the value of θ when the bob is released to t=0) and the angular frequency is given by
The period (the time for one complete oscillation) is then:
QUESTIONS
1. Does the period (T) depend on the mass (m), the amplitude A (θmax) and the length (L) of a pendulum?
The period (T) does not depend on neither the mass nor the amplitude however, it does depend on the length of a pendulum.
2. Suppose that you did this experiment on the moon. What would its period be there? (The acceleration due to gravity on the moon is 1.6 m/s2).
The difference in period would depend on the length of the pendulum. For example, if the length is set to 1.460 m, the equation could be used in the following manner:
and the period would equal 6 seconds. The
PURPOSE The purpose of this experiment is to study how the period of a pendulum depends on length, mass, and amplitude of the swing.
THEORY A simple pendulum is an idealized model consisting of a point mass (sometimes called a pendulum bob) suspended by a massless unstretchable string. When the bob is pulled to one side of its straight down equilibrium position and released, it oscillates about the equilibrium position. The path of the bob is not a straight line but an arc of a circle with radius L equal to the length of the string. We use as our coordinate the distance x measured along the arc. If the motion is simple harmonic, the restoring force must be directly proportional to x or to θ because x=L θ. The restoring force F is the tangential component of the gravity. The restoring force is proportional not to θ but sin θ, so the motion is not simple harmonic. However, if the angle θ is small, sin θ is very nearly equal to θ (in radians). Thus we have:
The bob will then execute simple harmonic motion and x (or θ) will be a sinusoidal function of time:
Where A, called the amplitude, is the maximum value of θ (i.e., A=θmax, or the value of θ when the bob is released to t=0) and the angular frequency is given by
The period (the time for one complete oscillation) is then:
QUESTIONS
1. Does the period (T) depend on the mass (m), the amplitude A (θmax) and the length (L) of a pendulum?
The period (T) does not depend on neither the mass nor the amplitude however, it does depend on the length of a pendulum.
2. Suppose that you did this experiment on the moon. What would its period be there? (The acceleration due to gravity on the moon is 1.6 m/s2).
The difference in period would depend on the length of the pendulum. For example, if the length is set to 1.460 m, the equation could be used in the following manner:
and the period would equal 6 seconds. The