**Description:**

A look at the physics that govern the pendulum.

**Watch Video:**

**Other Demos of Interest:**

**Possible Incorporated Topics:**

- oscillatory motion
- conservation of energy
- free-body diagram

**Theory**:

A pendulum consists of a mass suspended from a string that is fixed to a pivot. The mass is free to swing back and forth, and rests at what we will call the “equilibrium position”.

We will say that the string has length *L* and the ball suspended from it has mass *m*. For simplicity, we will assume that the pivot is frictionless and the string is massless. The displacement of the pendulum from equilibrium can be described by an angle *θ* and an arc-length *s*. Both of these parameters are 0 when the pendulum is at the equilibrium position.

By convention, angles are measured such that counterclockwise is positive and clockwise is negative. When the pendulum swings to the right, both *θ *and *s* are positive, while they are both negative when the pendulum swings to the left.

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We can construct a free-body diagram to examine the forces acting on the pendulum mid-swing:

We can define the axes to be such that the y-axis lies along the string and the x-axis is perpendicular to it. The string provides a tension force (*T*) on the ball, which acts in the positive y-direction. The only other force acting on the ball is the force of gravity (*F _{g}*). This force acts in the downward direction, and can be broken up into its x- and y-components (

*F*and

_{g}^{x}*F*). Since

_{g}^{y}*F*and

_{g}^{y}*T*are of equal magnitude and directed in opposite directions, those forces cancel one another so there is no net force in the y-direction. The net force thus consists only of

*F*:

_{g}^{x}The force of gravity is expressed as:

so the x-component can be written as:

Newton’s second law is:

so our equation of motion for the pendulum is thus:

Writing acceleration as the second derivative of distance (*s*), this becomes:

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The pendulum moves a distance along an arc length (*s*) of a circle with radius r:

We can express the arc length as:

(with *θ* in radians).

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To simplify the equation of motion, we can implement the small-angle approximation, which holds for angles less than ~10°. We can look at the right-angle triangle created by the ball with the equilibrium position, denoting the horizontal distance from the ball to the equilibrium as *h*. Since this is a right-angle triangle, we can trigonometrically express h as:

If *θ *is small, the lengths of *h* and *s* will be approximately the same:

and therefore:

which gives rise to the small angle approximation:

(where *θ* is in radians).

From this it follows that:

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Using the small angle approximation, we can simplify our equation of motion to:

Replacing *θ *using the definition of arc length (*s = r **θ*) and using the approximation that *r ~ L*, our equation becomes:

This is a second-order differential equation, for which the solution is:

where *A* is the amplitude of the oscillation and φ_{0} the phase, both of which are set by the initial conditions of the pendulum. The angular frequency, *ω*, is defined as:

where *f* is the frequency and *T* the period. This solution yields an angular frequency of:

for the pendulum. The period and frequency of the pendulum do NOT depend on the mass of the pendulum, they depend ONLY on the length of the string. A 1 gram pendulum will have the same period as a 100 kg pendulum so long as the strings are the same length.

Rearranging the above equations, we see that the the period of a pendulum (that is, the time to complete one full swing back and forth) is given by:

which depends only on the length of the string.

**Apparatus:**

- two pendulums of different masses (cork and metal balls were used here), with adjustable length strings
- stopwatch
- metre stick

**Procedure:**

- hang two pendulums of different masses from strings the same length (after measuring with metre stick)
- using the stopwatch, measure the period of each pendulum and note that they are the same, independent of their mass
- change the length of the string of one of the pendulums and note how the period changes