Question

A man of mass \(100 .\) kg feels a gravitational force, \(F_{\mathrm{m}}\), from a woman of mass \(50.0 \mathrm{~kg}\) sitting \(1 \mathrm{~m}\) away. The gravitational force, \(F_{w}\) experienced by the woman will be ___________ that experienced by the man. a) more than b) less than c) the same as d) not enough information given

Short answer

Answer: c) the same as Explanation: According to Newton's law of gravitation, the gravitational force between the man and woman is equal and opposite, following Newton's third law of motion. Therefore, the gravitational force experienced by the woman will be the same as that experienced by the man.

Step-by-step solution

Understand Newton's law of gravitation

Newton's law of gravitation states that the force of gravity acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula for the gravitational force is: \(\textrm{Force}=G \times \frac{m_1 \times m_2}{r^2}\) where \(\textrm{Force}\) is the gravitational force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses of the objects and \(r\) is the distance between them.

Calculate the gravitational force acting on the man, Fm

Using the information provided in the problem, we have: \(F_{m} = G \times \frac{m_{\mathrm{man}} \times m_{\mathrm{woman}}}{r_{\mathrm{mw}}^2}\) where \(F_{m}\) is the gravitational force on the man, \(m_{\mathrm{man}} = 100 \mathrm{~kg}\), \(m_{\mathrm{woman}} = 50 \mathrm{~kg}\), and \(r_{\mathrm{mw}} = 1 \mathrm{~m}\).

Calculate the gravitational force acting on the woman, Fw

Similarly, we can calculate the gravitational force acting on the woman: \(F_{w} = G \times \frac{m_{\mathrm{woman}} \times m_{\mathrm{man}}}{r_{\mathrm{wm}}^2}\) where \(F_{w}\) is the gravitational force on the woman, \(m_{\mathrm{woman}} = 50 \mathrm{~kg}\), \(m_{\mathrm{man}} = 100 \mathrm{~kg}\), and \(r_{\mathrm{wm}} = 1 \mathrm{~m}\).

Compare Fm and Fw

Notice that the two equations in Step 2 and Step 3 are the same, with only \(m_{\mathrm{man}}\) and \(m_{\mathrm{woman}}\) switched. This shows that the forces experienced by both individuals are equal and opposite, following Newton's third law of motion. So, the gravitational force experienced by the woman, \(F_{w}\), will be the same as that experienced by the man, \(F_{m}\). Therefore, the correct answer is: c) the same as

Key concepts

Newton's Law of Gravitation

Imagine two people in space, held in place by an unseen bond. This is akin to gravity—a force that acts unseen between two masses. Sir Isaac Newton's famed law of gravitation is the invisible thread that tells us precisely how this force behaves. It states that every particle attracts every other particle with a force that is directly proportional to the product of their masses. This force is also inversely proportional to the square of the distance that separates their centers.

Put simply, if you double the mass of one object, the gravitational force between it and another object doubles too. If you move them twice as far apart, the force drops by a factor of four (since the distance squared is four). This law applies universally, whether explaining how an apple falls from a tree or how the moon orbits the Earth.

In the case of our exercise, a man and a woman, each with their own mass, exert a gravitational pull on one another. Newton's law enables us to quantify this interaction and understand that the force they both feel relies on both of their masses as well as the distance between them.

Gravitational Constant

Now that we've framed Newton's law of gravitation conceptually, let's inject some precision with the gravitational constant, represented by the symbol 'G'. This is the glue that brings our understanding of gravitational force into the realm of calculable reality.

The gravitational constant is a proportionality factor used in the equation for Newton's law of gravitation and has a fixed value of approximately 6.674×10^-11 N·(m/kg)². It is remarkably small, indicating that the force of gravity is weak compared to other fundamental forces.

However, despite being comparatively weak, gravity has an infinite range and affects every object with mass, allowing it to shape the cosmos. The gravitational constant ensures that we can calculate the precise attraction between any two masses, regardless of whether they are as vast as planets or as diminutive as the man and woman in our exercise.

Inverse Square Law

Contained within Newton's law of gravitation is a fundamental principle known as the 'inverse square law'. This principle predicts that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. The 'square' part refers to the fact that you square the distance, multiplying it by itself.

In the context of gravity, this means that if you double the distance between two objects, the gravitational force is not just halved—it reduces to one-quarter of its original value. This is because of the square of two (the change in distance) is four. It's a critical concept for understanding why celestial bodies interact the way they do and can help us make sense of the delicate balance within our solar system.

For our textbook exercise, the inverse square law is integral in comprehending why the gravitational forces experienced by the man and the woman remain the same. Even though their masses are different, since the distance remains constant, any changes on one side of the formula are perfectly balanced by equal changes on the other.