Question

Newton was holding an apple of mass \(100 . \mathrm{g}\) and thinking about the gravitational forces exerted on the apple by himself and by the Sun. Calculate the magnitude of the gravitational force acting on the apple due to (a) Newton, (b) the Sun, and (c) the Earth, assuming that the distance from the apple to Newton's center of mass is \(50.0 \mathrm{~cm}\) and Newton's mass is \(80.0 \mathrm{~kg}\).

Short answer

#tag_title# Question Calculate the magnitudes of the gravitational forces acting on a 100g apple due to Newton, the Sun, and the Earth.

Step-by-step solution

Calculate the gravitational force due to Newton

F_N = G * (m1 * m_N) / d_N^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 80.0kg) / (0.5m)^2. ## Part (b): Gravitational force due to the Sun ##

Calculate the gravitational force due to the Sun

F_s = G * (m1 * m_s) / d_s^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 1.989 x 10^30 kg) / (1.496 x 10^11 m)^2. ## Part (c): Gravitational force due to the Earth ##

Calculate the gravitational force due to the Earth

F_E = G * (m1 * m_E) / d_E^2 = (6.674 x 10^-11 N⋅m²/kg²) * (0.1kg * 5.972 x 10^24 kg) / (6.371 x 10^6 m)^2. After calculating the gravitational force for each case using the given values and the formula, we get the magnitudes of the gravitational forces acting on the apple due to Newton, the Sun, and the Earth.

Key concepts

Gravitational Force

Gravitational force is a fundamental interaction through which massive objects attract one another. It is the force that the apple in the exercise experiences due to the presence of other massive bodies. Based on Newton's Law of Universal Gravitation, the gravitational force can be calculated using the formula:

• \[ F = G \frac{{m_1 m_2}}{r^2} \]

where:

• \( F \) is the gravitational force,
• \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \text{ Nm}^2/ ext{kg}^2 \),
• \( m_1 \) and \( m_2 \) are the masses of the two objects,
• \( r \) is the distance between the centers of the two masses.

Gravitational force is always attractive, pulling the two masses towards each other. It plays a crucial role in forming and maintaining the orbits of planets and moons, among many other natural phenomena. In the exercise, the gravitational force acting on the apple was calculated with respect to various celestial and human bodies: Newton, the Sun, and Earth.

Mass and Weight

In physics, mass and weight are related but distinct concepts. Mass is a measure of the amount of matter within an object, and it remains constant regardless of the object's location. It is usually measured in kilograms. For example, in the exercise, the apple's mass was given as \( 0.1 \text{ kg} \) or 100 grams.
Weight, on the other hand, refers to the gravitational force acting on an object. It varies depending on the gravitational pull of the planet or body on which the object resides. The weight of an object can be calculated using the formula:

• \[ W = m \cdot g \]

where:

• \( W \) is the weight,
• \( m \) is the mass,
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \text{ m/s}^2 \) on Earth.

Thus, while the mass of the apple remains \( 0.1 \text{ kg} \), its weight would be different depending on the gravitational forces exerted by the Earth or any other celestial body.

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The equation for this gravitational attraction is:

• \[ F = G \frac{{m_1 m_2}}{r^2} \]

This law explains not only why and how objects attract each other but also how gravity operates at both terrestrial and cosmic scales. For example, this is why Earth can hold an atmosphere and why planets orbit around stars, such as our solar system does around the Sun.
To see this law in action, consider the calculations from the exercised problem:

• The gravitational pull on the apple by the Sun, though physically distant, is reliant on the Sun's massive size, showing how mass contributes to gravitational force.
• The short distance between the apple and Newton himself shows how even smaller bodies can exert a gravitational force, though it is nearly negligible compared to the Sun or Earth.
• The gravitational interaction with Earth is what chiefly contributes to the apple's weight and is noticeable in everyday life.

Thus, Newton's law provides a fundamental understanding of how and why gravitational forces behave the way they do.