Two uniform spheres, each with mass M and radius R, touch each other. What is the magnitude of their gravitational force of attraction?

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Identify the formula for the gravitational force between two masses: \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between the centers of the two masses.

Since the spheres are touching each other, the distance \( r \) between their centers is equal to the sum of their radii, which is \( 2R \).

Substitute the given values into the formula: \( F = \frac{G \cdot M \cdot M}{(2R)^2} \).

Simplify the expression: \( F = \frac{G \cdot M^2}{4R^2} \).

This expression gives the magnitude of the gravitational force of attraction between the two spheres.

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주요 개념

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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 their centers. The formula is F = G * (m1 * m2) / r^2, where G is the gravitational constant.

Center of Mass

The center of mass of an object is the point at which the mass of the object is considered to be concentrated for the purpose of analyzing translational motion. For uniform spheres, the center of mass is at the geometric center. When calculating gravitational force between two spheres, the distance between their centers of mass is used.

Gravitational Constant (G)

The gravitational constant, denoted as G, is a fundamental constant in physics that quantifies the strength of the gravitational force between two masses. Its value is approximately 6.674 × 10^-11 N(m/kg)^2. It is crucial for calculating gravitational forces in Newton's Law of Universal Gravitation.