Pendulum

The pendulum is a scientific tool that has been used for centuries to study the laws of physics and mechanics. The main characteristic of a simple pendulum is that, for small angles, the period of oscillation depends only on the length of the string and gravity.

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How to measure it in class

Measuring the period of a pendulum by hand is not very precise. FizziQ makes it possible to time it automatically, either from the sound of its passage past an obstacle, or by attaching the smartphone itself to the mass and using its accelerometer.

Steps:

Suspend a mass from a thin, inextensible string, and measure the length L from the suspension point to the center of the mass
Pull the pendulum aside by a small angle, 10° at most, and release it without initial velocity
Record the acceleration, or the sound produced at each passage, for about twenty oscillations
Measure the total duration of 20 periods and divide by 20: the uncertainty on the period is thus divided by 20
Repeat for five different lengths, from about 20 cm to 1.5 m
Plot T² as a function of L: the resulting straight line has slope 4π²/g, from which g is obtained

Scientific activities on this topic

We offer several pendulum experiments to carry out with a smartphone:

Learn more

The main characteristic of a simple pendulum is that, for small angles, the period of oscillation depends only on the length of the string and gravity, and it is for this reason that the pendulum was an essential component of clocks for many centuries.

In science, simple pendulums are used to study the principles of gravity and dynamics. They notably allowed scientists to show that the acceleration of gravity was not constant.

The equation for the period of a simple pendulum for small-angle oscillations is given by:

T = 2π * sqrt(l/g)

where T represents the period in seconds, l the length of the string in meters, and g the acceleration due to gravity in meters per second squared.

This equation shows that the oscillation period of a simple pendulum depends on the length of the string and the acceleration due to gravity. The longer the string, the longer the period. Likewise, the greater the acceleration due to gravity, the shorter the period.

The Newton’s cradle is another type of pendulum used to study collisions and to verify the theorem of conservation of energy.

The Foucault pendulum is another example of the use of the pendulum in science. It is a pendulum that demonstrates the rotation of the Earth. The Foucault pendulum consists of a heavy mass suspended from a very long wire. When the pendulum is in motion, it stays in the same plane of vibration, but the Earth rotates beneath it, so the plane of vibration appears to rotate relative to the Earth. Léon Foucault gave a public demonstration at the Panthéon in 1851, with a 67 m wire. The apparent rotation of the oscillation plane is complete in 24 h only at the poles: it takes 24 h / sin(latitude), about 32 h in Paris, and it is zero at the equator.

Simple pendulum and physical pendulum: do not confuse them

The simple pendulum is an idealized model: a point mass at the end of a massless, inextensible string. It, and it alone, obeys T = 2π√(L/g). A real object oscillating about an axis is a physical pendulum: its mass is distributed, and its period involves its moment of inertia J and the distance d from the axis to the center of gravity, according to T = 2π√(J/(mgd)). A homogeneous ruler of length L pivoting at one end, for example, oscillates like a simple pendulum of length only 2L/3 - it is faster than a string of the same length. In practice, the simple pendulum model remains excellent as long as the suspended mass is small compared with the length of the string.

Isochronism is only approximate

This is the most often misunderstood point. The relation T = 2π√(L/g) is valid only for small oscillations, when sin θ can be replaced by θ in the equation of motion. As soon as the amplitude increases, the period increases too: about +0.2% at 10°, +1.7% at 30°, nearly +4% at 45°. The independence of the period from the amplitude, observed by Galileo at the beginning of the 17th century, is therefore an approximate property, excellent at small angles and false at large ones. It is precisely this defect that limited the precision of the first pendulum clocks and drove Huygens, as early as 1656, to look for suspensions correcting the deviation.

In the curriculum

The simple pendulum belongs to the upper years of high school physics: study of a mechanical oscillator, conversion between gravitational potential energy and kinetic energy, conservation of mechanical energy in the absence of friction. It is also a classic support for measuring g and for uncertainty analysis. The method of plotting T² = f(L) commonly gives g to within 1 or 2%, that is between 9.7 and 9.9 m·s⁻².

Formula

Period of a simple pendulum, for small oscillations only:

T = 2π × √(L / g)

First-order amplitude correction, for a release at angle θ₀:

T ≈ 2π√(L/g) × (1 + θ₀²/16)

Period of a physical pendulum, oscillating about a fixed axis:

T = 2π × √(J / (m g d))

Determination of g by linear regression:

T² = (4π² / g) × L

Conservation of mechanical energy, without friction, for a release at height h above the lowest point:

½ m v² = m g h, hence v = √(2gh) when passing through the vertical

where:

T: period, duration of one complete back-and-forth swing (s)
L: length of the string, from the suspension point to the center of the mass (m)
g: gravitational acceleration, about 9.81 m·s⁻² in France (m·s⁻²)
θ₀: angular amplitude of the release, in radians (rad)
J: moment of inertia about the axis of rotation (kg·m²)
m: mass of the pendulum (kg)
d: distance from the axis to the center of gravity (m)
v: speed at the lowest point (m·s⁻¹)

Application examples

One-meter pendulum. A 1.00 m string gives T = 2.006 s: a pendulum of about one meter beats almost exactly one second at each passage. The “seconds pendulum” of clocks, with a period of 2 s, measures precisely 0.994 m in Paris.
Measuring g. By plotting T² = f(L) over five lengths, a class commonly obtains g at 9.8 ± 0.2 m·s⁻². It is one of the few fundamental constants measurable with a string and a stopwatch.
Variation of g with latitude. In 1672, Jean Richer found that his pendulum clock, set in Paris, lost about two and a half minutes per day in Cayenne. The pendulum had to be shortened: g is weaker near the equator, because of the Earth’s rotation and flattening.
Foucault pendulum of the Panthéon. With its 67 m of wire, its period is about 16 s. Its length is not anecdotal: the longer and heavier the pendulum, the longer it oscillates without damping, which leaves time to see the oscillation plane rotate.
The swing of walking. The human leg oscillates like a physical pendulum: this is why the natural walking cadence is roughly fixed for a given person, and why shorter people walk with a higher step frequency.
Amplitude and error. Releasing a pendulum at 30° instead of 5° lengthens the period by 1.7%. On a measurement of g, this creates a systematic deviation of more than 3%, well above the experimental uncertainty: the error would look like an equipment problem when it actually comes from the protocol.

FAQ

Q: Does the period depend on the suspended mass? A: No, and this is the most surprising result of the simple pendulum. The mass does not appear in T = 2π√(L/g). A lead ball and a wooden ball at the end of the same string oscillate with the same period. The reason is the same as for free fall: the mass appears both in the weight and in the inertia, and cancels out.

Q: Why must the angles remain small? A: Because the formula relies on the approximation sin θ ≈ θ, valid only for small θ expressed in radians. Beyond that, the period increases: +1.7% at 30°, +4% at 45°. Below 10°, the deviation stays under 0.2%, hence negligible compared with classroom measurement uncertainties.

Q: What is the difference between a simple pendulum and a physical pendulum? A: The simple pendulum is a model: point mass, massless string. The physical pendulum is a real solid whose mass is distributed, and its period involves the moment of inertia. A massive object suspended from a short string behaves as a physical pendulum, not as a simple pendulum.

Q: Should L be measured to the bottom of the mass or to its center? A: To its center of gravity. This is the most frequent error in lab work, and it is systematic: it shifts all the points in the same direction and biases the value of g without degrading the alignment of the points, so it does not show up on the graph.

Q: Why does the pendulum eventually stop if energy is conserved? A: The total energy is indeed conserved, but the mechanical energy is not: friction from the air and at the attachment point gradually converts it into heat. The amplitude decreases, while the period remains practically unchanged as long as the damping is weak.

Harmonic Oscillator - Period - Mechanical Energy - Potential Energy - Kinetic Energy - Gravitational Acceleration - Newton’s Cradle - Resonance Frequency