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- Equation can be obtained using two definitions for torque.
First start by defining the torque on the pendulum bob using the force due to gravity.
where is the length vector of the pendulum and is the force due to gravity.
For now just consider the magnitude of the torque on the pendulum.
where is the mass of the pendulum, is the acceleration due to gravity, is the length of the pendulum, and is the angle between the length vector and the force due to gravity.
Next rewrite the angular momentum.
Again just consider the magnitude of the angular momentum.
and its time derivative
The magnitudes can then be compared using
thus:
which is the same result as obtained through force analysis. (en)
- thumb|right|alt=Coordinates of a simple gravity pendulum.|Coordinates of a simple gravity pendulum.|300px
can additionally be obtained through Lagrangian Mechanics. More specifically, using the Euler–Lagrange equations by identifying the Lagrangian of the system , the constraints and solving the following system of equations
If the origin of the Cartesian coordinate system is defined as the point of suspension , then the bob is at
and the velocity of the bob, calculated via differentiating the coordinates with respect to time
Thus, the Lagrangian is
The Euler-Lagrange equation is thus
Which can then be rearranged to match , obtained through force analysis.
Deriving via Lagrangian Mechanics, while excessive with a single pendulum, is useful for more complicated, chaotic systems, such as a double pendulum. (en)
- thumb|Figure 2. Trigonometry of a simple gravity pendulum.|right|300px
It can also be obtained via the conservation of mechanical energy principle: any object falling a vertical distance would acquire kinetic energy equal to that which it lost to the fall. In other words, gravitational potential energy is converted into kinetic energy. Change in potential energy is given by
The change in kinetic energy is given by
Since no energy is lost, the gain in one must be equal to the loss in the other
The change in velocity for a given change in height can be expressed as
Using the arc length formula above, this equation can be rewritten in terms of :
where is the vertical distance the pendulum fell. Look at Figure 2, which presents the trigonometry of a simple pendulum. If the pendulum starts its swing from some initial angle , then , the vertical distance from the screw, is given by
Similarly, when , then
Then is the difference of the two
In terms of gives
This equation is known as the first integral of motion, it gives the velocity in terms of the location and includes an integration constant related to the initial displacement . Next, differentiate by applying the chain rule, with respect to time to get the acceleration
which is the same result as obtained through force analysis. (en)
- thumb|Figure 1. Force diagram of a simple gravity pendulum.|right|300px
Consider Figure 1 on the right, which shows the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle is measured in radians, and this is crucial for this formula. The blue arrow is the gravitational force acting on the bob, and the violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion. The direction of the bob's instantaneous velocity always points along the red axis, which is considered the tangential axis because its direction is always tangent to the circle. Consider Newton's second law,
where is the sum of forces on the object, is mass, and is the acceleration. Newton's equation can be applied to the tangential axis only. This is because only changes in speed are of concern and the bob is forced to stay in a circular path. The short violet arrow represents the component of the gravitational force in the tangential axis, and trigonometry can be used to determine its magnitude. Thus,
where is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that and always point in opposite directions. This makes sense because when a pendulum swings further to the left, it is expected to accelerate back toward the right.
This linear acceleration along the red axis can be related to the change in angle by the arc length formulas; is arc length:
thus: (en)
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