Question

Newton's Law of Gravity specifies the magnitude of the interaction force between two point masses, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) as \(F(r)=G m_{1} m_{2} / r^{2} .\) The gravitational constant \(G\) can be determined by directly measuring the interaction force (gravitational attraction) between two sets of spheres by using the apparatus constructed in the late 18th century by the English scientist Henry Cavendish. This apparatus was a torsion balance consisting of a 6.00 -ft wooden rod suspended from a torsion wire, with a lead sphere having a diameter of 2.00 in and a weight of 1.61 lb attached to each end. Two 12.0 -in, 348 -lb lead balls were located near the smaller balls, about 9.00 in away, and held in place with a separate suspension system. Today's accepted value for \(G\) is \(6.674 \cdot 10^{-11} \mathrm{~m}^{3} \mathrm{~kg}^{-1} \mathrm{~s}^{-2}\) Determine the force of attraction between the larger and smaller balls that had to be measured by this balance. Compare this force to the weight of the small balls.

Short answer

Answer: In Cavendish's torsion balance experiment, the force of attraction between the large and small sphere is approximately 1.24 x 10^-7 N. This force is much weaker than the weight of the small sphere, which is 7.282 N, being around 5.93 x 10^-9 times as large.

Step-by-step solution

Convert the given measurements to SI units

First, we need to convert the given measurements from imperial to SI units: - Diameter of large sphere: 12.0 in = 0.3048 m - Diameter of small sphere: 2.00 in = 0.0508 m - Weight of small sphere: 1.61 lb = 7.282 N (by multiplying by 4.448 N/lb) - Mass of large sphere: 348 lb = 157.853 kg (by multiplying by 0.453592 kg/lb) - Distance between the centers of large and small spheres: 9.00 in + 0.5 * 2.00 in + 0.5 * 12.0 in = 11 in = 0.2794 m

Calculate the force of attraction using Newton's Law of Gravity

Using the mass of the large sphere, \(m_1 = 157.853 \ \mathrm{kg}\), and the distance between the centers of the spheres, \(r = 0.2794 \ \mathrm{m}\), we can determine the gravitational attraction force between them: \(F(r) = G \cdot \frac{m_1 \cdot m_2}{r^2}\) To find the mass of the small sphere, \(m_2\), we can use its weight, \(W = 7.282 \ \mathrm{N}\): \(W = m_2 \cdot g\) where \(g\) is the acceleration due to gravity, approximately \(9.81 \ \mathrm{m/s^2}\). We can find \(m_2\): \(m_2 = \frac{W}{g} = \frac{7.282 \ \mathrm{N}}{9.81 \ \mathrm{m/s^2}} = 0.743 \ \mathrm{kg}\) Now we can find the gravitational force \(F(r)\) between the two spheres: \(F(r) = (6.674 \times 10^{-11} \ \mathrm{m^3 kg^{-1} s^{-2}}) \cdot \frac{(157.853 \ \mathrm{kg}) \cdot (0.743 \ \mathrm{kg})}{(0.2794 \ \mathrm{m})^2} = 1.24 \times 10^{-7} \ \mathrm{N}\)

Comparing gravitational force with the weight of the small sphere

Finally, we will compare the gravitational force \(F(r)\) with the weight \(W\) of the small balls: Gravitational Force: \(F(r) = 1.24 \times 10^{-7} \ \mathrm{N}\) Weight of Small Sphere: \(W = 7.282 \ \mathrm{N}\) The gravitational force is much smaller than the weight of the small sphere; about \(5.93\times10^{-9}\) times as large to be more precise. The weight of the small sphere dominates its gravitational interaction with the large sphere in Cavendish's torsion balance experiment.

Key concepts

Gravitational Force Calculation

Understanding how to calculate gravitational force can be a bit daunting, but it's a fundamental concept in physics. When we calculate the gravitational force between two masses, we use Newton’s Universal Law of Gravitation, which states that every point mass attracts every other point mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The force of attraction, denoted as \( F \), between two masses \( m_1 \) and \( m_2 \) can be calculated using the formula:
\[ F(r) = G \cdot \frac{m_1 \cdot m_2}{r^2} \]
Here, \( G \) is the gravitational constant, and \( r \) is the distance between the centers of the two masses. This equation can give us an insight into not only the force that binds us to the Earth, but also the forces that govern the movement of celestial bodies like stars and planets.

Cavendish Experiment

The Cavendish Experiment, conducted in the late 18th century by Henry Cavendish, was a groundbreaking endeavor to measure the Earth's density, which in consequence also allowed for the determination of the gravitational constant, \( G \). The apparatus, known for its simplicity and ingenuity, consisted of a torsion balance with a bar suspended from a wire.

Apparatus and Methodology

At each end of the bar, a small lead sphere was attached, and it was placed near two larger fixed lead spheres. The gravitational attraction between the small and large spheres would cause the bar to twist, twisting the wire as well. By observing the amount of torsion, Cavendish calculated the force of attraction between the masses.

Because the gravitational forces in question were extremely small, the accuracy and sensitivity of the apparatus were critical. Cavendish's work was the first to yield accurate values for the gravitational constant and paved the way for the precise calculations of mass for Earth and other celestial bodies.

Gravitational Constant

The gravitational constant, denoted by \( G \), is a key component in the equation used to calculate gravitational force. Its value determines the strength of the gravitational force.

Defined as \( 6.674 \cdot 10^{-11} \, \mathrm{m}^3 \, \mathrm{kg}^{-1} \, \mathrm{s}^{-2} \), the gravitational constant is notoriously difficult to measure because the gravitational force is extremely weak compared to other fundamental forces. The value of \( G \) is critical in calculations ranging from spacecraft trajectories to predictions of planetary motions.

Importance in Astrophysics

\( G \) is also essential in astrophysics, particularly in understanding stellar and galactic behaviors. Without precise measurements of \( G \), much of our understanding of the universe's dynamics would lack the accuracy needed to make reliable predictions and understandings of cosmic phenomena. Regardless of the scale, the gravitational constant \( G \) remains a cornerstone in the formulas that allow us to comprehend gravity's role throughout the universe.