Newton's Law of Gravity: Videos & Practice Problems

The universal law of gravitation, formulated by Newton, states that all objects with mass attract each other through gravitational forces. The gravitational force (F_G) can be calculated using the equation FG = G(m₁⁢m₂) / r2 , where G is the universal gravitational constant (6.67 × 10⁻¹¹ N m² / kg² ) and r is the distance between the centers of mass. Understanding this principle is crucial for calculating gravitational forces in various contexts, including satellites orbiting planets.

Universal Law of Gravitation

The universal law of gravitation, formulated by Isaac Newton, states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This relationship can be expressed mathematically with the equation:

F_G = \frac{G \cdot m_1 \cdot m_2}{r^2}

In this equation, F_G represents the gravitational force between two masses, G is the universal gravitational constant, approximately equal to 6.67 × 10^-11 N m²/kg², m₁ and m₂ are the masses of the two objects, and r is the distance between their centers of mass.

Newton's law highlights that gravitational forces act along the line connecting the centers of the two masses, meaning that both objects exert equal and opposite forces on each other due to Newton's third law of motion. This principle is crucial in understanding not only terrestrial phenomena, such as an apple falling from a tree, but also celestial mechanics, like the orbits of planets and moons.

It is important to distinguish between the universal gravitational constant G and the local acceleration due to gravity g, which is approximately 9.8 m/s² on the surface of the Earth. While G remains constant throughout the universe, g varies depending on location, such as altitude or planetary body.

To illustrate the application of this law, consider two spheres, each with a mass of 30 kg, separated by a distance of 5 meters. To find the gravitational force between them, we substitute the values into the equation:

F_G = \frac{(6.67 \times 10^{-11}) \cdot (30) \cdot (30)}{(5)^2}

Calculating this gives:

F_G = \frac{(6.67 \times 10^{-11}) \cdot 900}{25} = 2.4 \times 10^{-9} N

This result indicates that the gravitational force between the two spheres is 2.4 × 10^-9 N, a minuscule force compared to everyday experiences, yet significant in the context of celestial bodies and their interactions.

 Two spheres are separated by 10m. If the lighter 40kg sphere feels a gravitational force of 1.6 × 10^-9 N, what is the mass of the heavier sphere?

• A. 6 kg
• B. 9.6 kg
• C. 60 kg
• D. 10 kg

Answer: (C)

Where is Net Force on middle object zero?

In this problem, we explore the gravitational forces between two spheres with known masses and a specified distance between them. We have a 10 kg mass (mass A) and a 25 kg mass (mass B) positioned 5 meters apart. The goal is to find the position of a third mass (mass C) along the line between them such that the net gravitational force acting on mass C is zero.

According to Newton's law of universal gravitation, the gravitational force between two masses is given by the formula:

F = \frac{G \cdot m_1 \cdot m_2}{r^2}

where F is the gravitational force, G is the gravitational constant, m₁ and m₂ are the masses, and r is the distance between the centers of the two masses. In our case, we denote the forces acting on mass C from mass A as F_AC and from mass B as F_BC.

To achieve equilibrium, we set the magnitudes of these forces equal:

F_{AC} = F_{BC}

Substituting the gravitational force equations, we have:

\frac{G \cdot m_A \cdot m_C}{r_A^2} = \frac{G \cdot m_B \cdot m_C}{r_B^2}

Since G and m_C appear on both sides, they can be canceled out, simplifying our equation to:

\frac{m_A}{r_A^2} = \frac{m_B}{r_B^2}

Next, we need to express r_B in terms of r_A. Given that the total distance between the two masses is 5 meters, we can write:

r_A + r_B = 5

From this, we can express r_B as:

r_B = 5 - r_A

Substituting this into our earlier equation gives us:

\frac{m_A}{r_A^2} = \frac{m_B}{(5 - r_A)^2}

Now, substituting the known masses (10 kg for mass A and 25 kg for mass B) leads to:

\frac{10}{r_A^2} = \frac{25}{(5 - r_A)^2}

To eliminate the fractions, we can cross-multiply:

10(5 - r_A)^2 = 25r_A^2

Expanding and simplifying this equation allows us to isolate r_A. After some algebraic manipulation, we find:

5 - r_A = 1.58r_A

Rearranging gives:

5 = 2.58r_A

Solving for r_A yields:

r_A = \frac{5}{2.58} \approx 1.94 \text{ meters}

This result indicates that mass C must be positioned approximately 1.94 meters from mass A. Consequently, mass C will be located about 3.06 meters from mass B, ensuring that the gravitational forces acting on it are balanced. This placement is closer to the lighter mass (mass A) due to the greater gravitational pull exerted by the heavier mass (mass B).

Two spheres of mass 300 kg and 500 kg are placed in a line 20 cm apart. If another sphere of mass 200 kg is placed between them, 8 cm from the 300 kg sphere, what is the net gravitational force on the 200 kg sphere?

• A. -7.32×10^-4 N (toward 300kg sphere)
• B. -1.62×10^-4 N (toward 300kg sphere)
• C. 7.32×10^-4 N (toward 500kg sphere)
• D. 1.62×10^-4 N (toward 500kg sphere)

Answer: (B)

Center of Mass Distance

The universal law of gravitation allows us to calculate the gravitational force between two objects, which can be represented as point masses. The formula for this force is given by:

F_g = \frac{G m_1 m_2}{r^2}

In this equation, F_g is the gravitational force, G is the gravitational constant, m₁ and m₂ are the masses of the two objects, and r is the distance between their centers of mass. When dealing with large objects, such as a planet and a satellite, we denote the mass of the planet as M and the mass of the satellite as m. The formula remains the same, but we now express it as:

F_g = \frac{G M m}{r^2}

In this context, r represents the distance from the center of the planet to the satellite, which can be expressed as the sum of the planet's radius R and the height h of the satellite above the planet's surface:

r = R + h

Here, R is a constant (the radius of the Earth), while h is a variable that can change depending on the satellite's altitude. To find the height h when given a specific gravitational force acting on the satellite, we can rearrange the gravitational force equation to solve for r first:

r = \sqrt{\frac{G M m}{F_g}}

Once we have calculated r, we can then find h by substituting back into the equation:

h = r - R

For example, if we want to determine the height at which the gravitational force on a satellite is 1000 newtons, and the mass of the satellite is 1000 kilograms, we can use the known values for G (approximately 6.67 × 10^-11 N m²/kg²), the mass of the Earth (approximately 5.97 × 10²⁴ kg), and the radius of the Earth (approximately 6.37 × 10⁶ m) to find r.

After calculating r, we can then find h by subtracting the radius of the Earth from the total distance r. This process ultimately leads to the height above the Earth's surface where the gravitational force on the satellite equals 1000 newtons, which in this case is approximately 13,600 kilometers.

Find force between spheres with radii

In this scenario, we analyze the gravitational attraction between two identical spheres, each with a mass of 10 kilograms and a diameter of 60 centimeters (0.6 meters). When the surfaces of these spheres are just barely in contact, we need to determine the gravitational force acting between them using Newton's law of gravitation.

The formula for gravitational force is given by:

$$ F_g = \frac{G \cdot m_1 \cdot m_2}{r^2} $$

where:

• $$ F_g $$ is the gravitational force between the two masses,
• $$ G $$ is the gravitational constant, approximately $$ 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $$,
• $$ m_1 $$ and $$ m_2 $$ are the masses of the two spheres, and
• $$ r $$ is the distance between the centers of mass of the two spheres.

Since the spheres are identical, the distance between their centers of mass is equal to the sum of their radii. Each sphere has a radius of 0.3 meters (half of the diameter), so the center of mass distance, $$ r $$, is:

$$ r = 2 \times \text{radius} = 2 \times 0.3 \, \text{m} = 0.6 \, \text{m} $$

Now, substituting the values into the gravitational force equation:

$$ F_g = \frac{(6.67 \times 10^{-11}) \cdot (10) \cdot (10)}{(0.6)^2} $$

Calculating this gives:

$$ F_g = \frac{6.67 \times 10^{-11} \cdot 100}{0.36} $$

$$ F_g \approx 1.85 \times 10^{-8} \, \text{N} $$

This result indicates the gravitational force of attraction between the two spheres is approximately $$ 1.85 \times 10^{-8} $$ newtons. Understanding this concept is crucial as it illustrates the principles of gravitational interaction and the application of Newton's law of gravitation in real-world scenarios.

A 2,000-kg spacecraft is blasting away from the surface of an unknown planet the same size as the Earth. At 1500km above the surface, an instrument onboard reads the gravitational force to be 18000 N. What is the planet's mass?

• A. 8.36×10²⁴kg
• B. 2.89×10¹²kg
• C. 1.67×10²⁸kg
• D. 5.78×10¹⁵kg

Answer: (A)

Here’s what students ask on this topic:

What is Newton's Law of Universal Gravitation?

Newton's Law of Universal Gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is given by:

$F_G=G\frac{m_1{m}_2}{r^2}$

where $G$ is the universal gravitational constant (6.67 × 10⁻¹¹ N m² / kg²), $m_1$ and $m_2$ are the masses of the objects, and $r$ is the distance between their centers of mass.

How do you calculate the gravitational force between two objects?

To calculate the gravitational force between two objects, use Newton's Law of Universal Gravitation:

$F_G=G\frac{m_1{m}_2}{r^2}$

Here, $G$ is the universal gravitational constant (6.67 × 10⁻¹¹ N m² / kg²), $m_1$ and $m_2$ are the masses of the objects, and $r$ is the distance between their centers of mass. Plug in the values for the masses and the distance to find the gravitational force.

What is the difference between the universal gravitational constant (G) and the local gravitational constant (g)?

The universal gravitational constant, $G$, is a constant value of 6.67 × 10⁻¹¹ N m² / kg² that applies everywhere in the universe. It is used in Newton's Law of Universal Gravitation. The local gravitational constant, $g$, is approximately 9.8 m/s² and represents the acceleration due to gravity at the Earth's surface. While $G$ is universal, $g$ can vary depending on the location, such as on different planets or altitudes.

How does the distance between two objects affect the gravitational force between them?

The gravitational force between two objects is inversely proportional to the square of the distance between their centers of mass. This relationship is described by Newton's Law of Universal Gravitation:

$F_G=G\frac{m_1{m}_2}{r^2}$

As the distance $r$ increases, the gravitational force $F_G$ decreases rapidly. For example, if the distance between two objects is doubled, the gravitational force between them becomes one-fourth as strong.

What is the significance of the universal gravitational constant (G)?

The universal gravitational constant, $G$, is a fundamental constant in physics that quantifies the strength of the gravitational force between two masses. Its value is 6.67 × 10⁻¹¹ N m² / kg². This constant is crucial for calculating gravitational forces in various contexts, from the attraction between small objects to the gravitational pull between celestial bodies. It ensures that the gravitational force can be consistently calculated regardless of the masses involved or their separation distance.