Answer :
Final answer:
To determine the gravitational force on an apple by Newton, the Sun, and the Earth, we apply Newton's Law of Universal Gravitation. The force is a product of both masses, inversely proportional to the square of the distance between them. The calculations assume that the gravitational effects of the Earth and Sun can be represented as if their masses were concentrated at their centers, an assumption that is valid for spherical bodies with uniform mass distribution.
Explanation:
Gravitational Forces Acting on an Apple
To answer the student's question:
a) The gravitational force between Newton and the apple can be calculated using Newton's law of universal gravitation, which states that any two bodies attract each other with a force that is directly proportional to the masses of the bodies and inversely proportional to the square of the distance between their centers of mass. Mathematically, the force is given by the formula:
[tex]F = G * (m1 * m2) / r^2,[/tex] where:
- F is the gravitational force,
- G is the gravitational constant (approximately 6.674 x 10⁻¹¹ N(m/kg)²),
- m1 and m2 are the masses of the two objects,
- r is the distance between their centers of mass.
Plugging in the values, we get F = (6.674 x 10⁻¹¹ N(m/kg)²), * (71.5 kg * 0.261 kg) / [tex](0.517 m)^2[/tex], which after calculation gives us the force exerted on the apple by Newton.
b) To calculate the gravitational force exerted on the apple by the Sun, we use the same formula with the mass of the Sun and the average distance from the apple to the Sun.
c) The calculation of the gravitational force exerted on the apple by the Earth is similar, using Earth's mass and the average distance from the apple to Earth's center.
In a simplified model, despite the non-uniform distribution of Earth's density, we can consider all of its mass to be concentrated at its center for the purpose of calculating gravitational forces, as supported by Newton's shell theorem.
Q: Newton was holding an apple of mass 261 g and thinking about the gravitational forces exerted on the apple by himself and by the Sun.
a) Calculate the magnitude of the gravitational force acting on the apple due to Newton, assuming that the distance from the apple to Newton's center of mass is 51.7 cm and Newton's mass is 71.5 kg.
b) Calculate the magnitude of the gravitational force acting on the apple due to the Sun. The mass of the Sun is 1.989×10³⁰ kg, and the average distance from the apple to the Sun is 1.496×10¹¹ m.
c) Calculate the magnitude of the gravitational force acting on the apple due to the Earth. The mass of the Earth is 5.972×10²⁴ kg, and the average distance from the apple to the Earth is 6.37×10⁶ m (radius of the Earth).
