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In the standard derivation of pendulum characteristics, at least through high school

and undergraduate Physics, an approximation is always made that assumes a small

angular displacement.

With that assumption, the angular displacement doesn't appear in the formula for

the period, i.e. the period depends on the pendulum's effective length, and is

independent of the angular displacement.

Q: How does the period of a pendulum vary theoretically with angular displacement?

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The length of the pendulum is measured from the pendulum's point of suspension to the center of mass of its bob. Its amplitude is the string's angular displacement from its vertical or its equilibrium position.

For small swings, and a simple pendulum:T = 2 pi root(L/g) where T is the time for one period, L is the length of the pendulum, and g is the strength of the gravitational field.

The period of a simple pendulum is given by the formulaT = 2*pi*sqrt(L/g)where T = periodL = lengthand g = local acceleration due to gravity.Note that this formula is applicable only when the angular displacement of the pendulum is small. For a displacement of 22.5 degrees (a quarter of a right angle), the true period is approx 1% longer : a clock will lose more than 1/2 a minute every hour!

Actually, the period of a pendulum does depend slightly on the amplitude. But at low amplitudes, it almost doesn't depend on the amplitude at all. This is related to the fact that in such a case, the restoring force - the force that pulls the pendulum back to its center position - is proportional to the displacement. That is, if the pendulum moves away further, the restoring force will also be greater.

A longer pendulum has a longer period.

Related questions

The period of a pendulum is independent of the angular displacement (within small angles) and the mass of the ball. It is only dependent on the length of the pendulum and the acceleration due to gravity. This is known as the principle of isochronism of the pendulum, first discovered by Galileo.

The factors affecting a simple pendulum include the length of the string, the mass of the bob, the angle of displacement from the vertical, and the acceleration due to gravity. These factors influence the period of oscillation and the frequency of the pendulum's motion.

The length of the pendulum is measured from the pendulum's point of suspension to the center of mass of its bob. Its amplitude is the string's angular displacement from its vertical or its equilibrium position.

The physical parameters in the investigation of a simple pendulum include its length, mass of the bob, angle of displacement, gravitational acceleration, and the period of oscillation. By experimenting with these parameters, one can analyze the motion and behavior of the pendulum.

(a) directly with its square root.(b) not at all if it can be considered as a point mass which is significantly greater than the "string". Otherwise corrections are necessary. (c) not if the angle is small. Otherwise corrections are necessary.

The period is directly proportional to the square root of the length.

For small swings, and a simple pendulum:T = 2 pi root(L/g) where T is the time for one period, L is the length of the pendulum, and g is the strength of the gravitational field.

The period of a simple pendulum is given by the formulaT = 2*pi*sqrt(L/g)where T = periodL = lengthand g = local acceleration due to gravity.Note that this formula is applicable only when the angular displacement of the pendulum is small. For a displacement of 22.5 degrees (a quarter of a right angle), the true period is approx 1% longer : a clock will lose more than 1/2 a minute every hour!

(a) directly with its square root.(b) not at all if it can be considered as a point mass which is significantly greater than the "string". Otherwise corrections are necessary. (c) not if the angle is small. Otherwise corrections are necessary.

To illustrate the graph of a simple pendulum, you can plot the displacement (angle) of the pendulum on the x-axis and the corresponding period of oscillation on the y-axis. As the pendulum swings back and forth, you can record the angle and time taken for each oscillation to create the graph. The resulting graph will show the relationship between displacement and period for the simple pendulum.

The popular formula for the period of a pendulum works only for small angular displacements. In deriving it, you need to assume that theta, the angular displacement from the vertical, measured in radians, is equal to sin(theta). If not, you need to make much more complicated calculations. There are also other assumptions to simplify the formula - eg string is weightless. The swing of the pendulum will precess with the rotation of the earth. This may not work if the pendulum hits its stand! See Foucault's Pendulum (see link). The motion of the pendulum will die out as a result of air resistance. Thermal expansion can change the length of the pendulum and so its period.

With a simple pendulum, provided the angular displacement is less than pi/8 radians (22.5 degrees) it will be a straight line, through the origin, with a slope of 2*pi/sqrt(g) where g is the acceleration due to gravity ( = 9.8 mtres/sec^2, approx). For larger angular displacements the approximations used in the derivation of the formula no longer work and the error is over 1%.