Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

문제 39

13장, 문제 39

Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 x 10^-6 N when their centers are 20 cm apart. What is the mass of each?

검증된 단계별 안내

Start by recalling Newton's law of universal gravitation:

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

, where

F

is the gravitational force,

G

is the gravitational constant (

6.674 \(\times\) 10^{-11} \ \(\text{N·m}\)^2/\(\text{kg}\)^2

m_1

and

m_2

are the masses of the two objects, and

r

is the distance between their centers.

Substitute the known values into the formula:

8.00 \(\times\) 10^{-6} = \(\frac{(6.674 \times 10^{-11}\)) m_1 m_2}{(0.20)^2}

. Note that the distance

r

is converted to meters (20 cm = 0.20 m).

Simplify the equation to isolate the product of the masses:

m_1 m_2 = \(\frac{(8.00 \times 10^{-6}\)) (0.20)^2}{6.674 \(\times\) 10^{-11}}

. This gives the value of

m_1 m_2

Use the fact that the combined mass of the two objects is 150 kg:

m_1 + m_2 = 150

. This provides a second equation to solve for

m_1

and

m_2

Solve the system of equations:

m_1 m_2 = \(\text{value from step 3}\)

and

m_1 + m_2 = 150

. Use substitution or the quadratic formula to find the individual masses

m_1

and

m_2

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Law of Universal Gravitation

This law 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)/r², 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.

Mass and Weight

Mass is a measure of the amount of matter in an object, typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass. In this problem, the combined mass of the two objects is given, which is essential for determining the individual masses using the gravitational force between them.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate variables and solve for unknowns. In this context, it is necessary to express the masses of the two objects in terms of their combined mass and the gravitational force, allowing for the calculation of each mass using the provided information.