14.6
Beschleunigung durch die Schwerkraft auf der Erde
Nach Newtons Gravitationsgesetz ist die Gravitationskraft auf einen Körper proportional zu seiner Masse. Gemäß dem zweiten Newtonschen Bewegungsgesetz…
Ein Objekt gilt als frei fallend, wenn die einzige Kraft, die auf es wirkt, die Erdanziehungskraft ist. Die Beschleunigung eines solchen frei fallenden Objekts wird als Beschleunigung durch die Schwerkraft bezeichnet, die mit g bezeichnet wird.
Nach dem zweiten Newtonschen Bewegungsgesetz ist die Größe der Kraft, die auf das Objekt wirkt, gleich seiner Masse multipliziert mit seiner Beschleunigung. Es wird auch als Gewicht des Objekts bezeichnet.
Setzt man mg mit der Gravitationskraftgleichung gleich, so kann die durch die Schwerkraft verursachte Beschleunigung des Objekts als Produkt aus der Gravitationskonstante und der Erdmasse dividiert durch das Quadrat seiner Entfernung vom Erdmittelpunkt ausgedrückt werden.
Der durchschnittliche Messwert von g nahe der Erdoberfläche liegt bei 9,8 m/s2. Da g unabhängig von der Masse des Objekts ist, fallen alle Massen in der Nähe der Erdoberfläche mit der gleichen Beschleunigung frei ab.
Wenn man also die Werte der Erdbeschleunigung durch die Schwerkraft, die Gravitationskonstante und die Entfernung als Erdradius einsetzt, wird die Masse der Erde auf 5,97 × 1024 Kilogramm geschätzt.
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Q1: What is acceleration due to gravity and how is it defined?
Acceleration due to gravity, denoted by g, is the acceleration of an object when only Earth's gravitational force acts on it during free-fall. According to Newton's second law, this acceleration equals the gravitational constant multiplied by Earth's mass, divided by the square of the distance from Earth's center. The average measured value near Earth's surface is 9.8 m/s².
Q2: Why do all objects fall with the same acceleration regardless of their mass?
According to Newton's law of gravitation, gravitational force is proportional to an object's mass. However, Newton's second law shows acceleration is inversely proportional to mass. These effects cancel out, making acceleration independent of mass. Therefore, all objects near Earth's surface experience the same gravitational acceleration of 9.8 m/s².
Q3: How does Earth's rotation affect the acceleration due to gravity?
The acceleration due to gravity varies from the equator to the poles because of Earth's rotation about its axis. This rotational effect causes measurable differences in g at different latitudes. However, for most practical purposes near Earth's surface, such as on Mount Everest, these variations are negligible and g can be treated as constant.
Q4: How can Earth's mass be calculated from the acceleration due to gravity?
Since Earth's mass and average radius are related to the gravitational constant and acceleration due to gravity, one can be estimated if the other is known. By substituting measured values of g, the gravitational constant, and Earth's radius into the gravitational equation, Earth's mass is calculated to be approximately 5.97 × 10²⁴ kilograms.
Q5: Does the acceleration due to gravity change significantly at high altitudes?
Near Earth's surface, g remains approximately constant at 9.8 m/s². However, at distances hundreds of kilometers above Earth's surface, the value of g becomes considerably different. This occurs because g depends on the square of the distance from Earth's center, so greater altitudes produce measurable reductions in gravitational acceleration.
Q6: What is the relationship between weight and acceleration due to gravity?
An object's weight is the gravitational force acting on it, calculated as mass times acceleration due to gravity (mg). According to Newton's second law, this force equals mass multiplied by acceleration. Since g is independent of an object's mass, weight is directly proportional to mass, with g serving as the proportionality constant.
Q7: Why is Earth considered an inertial frame of reference for studying gravity?
Although every object applies an equal and opposite gravitational pull on Earth according to Newton's third law, Earth's acceleration is negligible because its mass is vastly larger than ordinary objects near it. This makes Earth an appropriate inertial frame of reference for studying the dynamics of objects placed on or near its surface.