Ch 13: Newton's Theory of Gravity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 13: Newton's Theory of Gravity

Problem 4

Chapter 13, Problem 4

The International Space Station orbits 300 km above the surface of the earth. What is the gravitational force on a 1.0 kg sphere inside the International Space Station?

Verified step by step guidance

Step 1: Understand the problem. The gravitational force on the sphere is determined by Newton's law of universal gravitation, which states that the force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Step 2: Write down the formula for gravitational force:

F = G * (m₁ * m₂) / r²

, where

G

is the gravitational constant (

6.674 × 10⁻¹¹ N·m²/kg²

m₁

is the mass of the Earth,

m₂

is the mass of the sphere, and

r

is the distance between the center of the Earth and the sphere.

Step 3: Determine the values for the variables. The mass of the Earth is approximately

5.972 × 10²⁴ kg

. The radius of the Earth is approximately

6,371 km

, and the sphere is located

300 km

above the Earth's surface. Therefore, the total distance

r

6,371 km + 300 km = 6,671 km

, which must be converted to meters:

r = 6,671 × 10³ m

Step 4: Substitute the values into the formula. Replace

G

m₁

m₂

, and

r

with their respective values:

F = (6.674 × 10⁻¹¹) * (5.972 × 10²⁴) * (1.0) / (6,671 × 10³)²

Step 5: Simplify the expression to calculate the gravitational force. Perform the arithmetic operations step by step, ensuring proper handling of exponents and units. The result will give the gravitational force in newtons (N).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. This force is what keeps objects in orbit, including the International Space Station (ISS) around the Earth.

Weightlessness in Orbit

Weightlessness, or microgravity, occurs when an object is in free fall, such as when the ISS orbits the Earth. Although gravity is still present, the ISS and everything inside it, including a 1.0 kg sphere, are falling towards Earth at the same rate, creating the sensation of weightlessness. This condition affects how gravitational force is perceived by objects inside the ISS.

Altitude and Gravitational Acceleration

Gravitational acceleration decreases with altitude, but it does not become zero. At 300 km above Earth's surface, the gravitational acceleration is approximately 9.1 m/s², compared to 9.81 m/s² at sea level. This reduction is important for calculating the gravitational force acting on objects in the ISS, as it directly influences the force experienced by the 1.0 kg sphere.