Monday, February 25, 2019

Unizor - Physics4Teens - Energy - Potential Energy - Introduction

Notes to a video lecture on

Potential Energy - Introduction

Potential energy is a quantitative characteristic of an object's position relative to other objects or environment that are a source of some forces. These forces would result in motion of the object under consideration (if no counteracting forces present) and, therefore, would result in performing certain work related to this motion.

Typical example of potential energy is related to a position of some object in the gravitational field.
Since a planet Earth attracts all objects, an object raised to a certain height above the ground will fall, if no support is provided. The motion of falling object is the result of work of the forces of gravity. Therefore, by definition, an object raised above the ground has certain amount of potential energyequal to work performed by the forces of gravity to move it down to the ground.

In a simplified case of a small object of mass m, raised on small (relatively to the size of the Earth) height h, the force of gravity, acting on this object, is constant and is equal to the object's weight P = m·g, where g ≅ 9.8m/sec² is the acceleration of the free fall near the Earth's surface.
If no support is provided, the object will fall straight down. Thus, the constant force of gravity P would perform work, pushing the object down at a distance h, thus performing work
W = P·h = m·g·h

Let's examine the relationship between an object's potential energy and its kinetic energy as it falls down from certain initial height.
Assume an inertial frame of reference associated with Earth and X-coordinate directed straight up perpendicularly to the ground with ground level be at coordinate X=0.
Initial position of the object is, therefore, at X-coordinate equal to X=h. The falling object moves with constant acceleration −g (negative, since it moves from positive X=h to X=0). Initial speed of the object is zero.
Therefore, the time-dependent motion X(t) can be described by the kinematic equation
X(t) = h − g·t²/2
The object's velocity is directed vertically down and, as a function of time, is the first derivative of the X(t):
V(t) = X'(t) = −g·t
(which is obvious from the fact that the motion is with constant acceleration −g and zero initial speed)

As object falls down, its height diminishes, which means that its potential energy also diminishes. At any given moment of time t this energy equals to
Epot(t) = m·g·X(t) = m·g·(h−g·t²/2)

At the same time the kinetic energy of the object is increasing, since its speed V(t)is increasing. At any given moment of time t this energy equals to
Ekin(t) = m·V²(t)/2 = m·g²·t²/2

Remarkably, the full mechanical energy of this object, which is a sum of its potential energy and kinetic energy is constant:
Efull = Epot + Ekin = m·g·h,
which is its potential energy in the beginning of motion (when its initial speed and, therefore, kinetic energy are zero).

The total time in motion from the initial position above the ground at height h to the ground can be obtained by resolving the equation X(t)=0 for variable t:
X(t) = h − g·t²/2 = 0
tfall = √2h/g
At that time of touching the ground the potential energy of an object equals to zero and its kinetic energy equals to
Ekin(tfall) = m·V²(tfall)/2 = m·g²t²fall/2 = m·g·h

In the beginning the potential energy is at its maximum and kinetic energy at zero. At the end the potential energy is at zero, while kinetic energy at its maximum, but their sum is always the same. So, it appears that, as an object falls, its potential energy is transformed into kinetic energy, preserving their sum - the full mechanical energy.

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