Problem 14 - Calculus of a Single Variable

Question

Propulsion \(A\) lunar module weighs 12 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of the moon to be 1100 miles and its force of gravity to be one-sixth that of Earth.

Step-by-Step Solution

Answer

The work done on the lunar module is calculated by first converting its weight to mass, then calculating the force of gravity it experiences on the moon, and finally multiplying this force by the distance the module is propelled. Implementing these calculations will yield the desired result.

Step 1: Convert the weight of the lunar module to mass

Since weight is the force exerted by gravity on an object's mass, one can calculate the lunar module's mass by dividing its weight by Earth's gravitational acceleration. Using the formula \[ m = \frac{w}{g} \], where \(m\) is the mass, \(w\) is the weight (12,000 kg since 1 ton equals 1,000 kg) and \(g\) is Earth's gravitational acceleration (9.81 m/s²). Hence, assuming Earth’s gravity, the mass of the module is \[ m = \frac{12,000 kg}{9.81 m/s²} \]

Step 2: Compute the force of the moon's gravity on the lunar module

The force of gravity on the moon can be calculated by multiplying the lunar module's mass by the moon's gravity, which is one-sixth of Earth's gravity. So, using the formula \[ F = m \cdot g_{moon} \], where \( F \) is the force, \( m \) is the mass calculated in step 1 and \( g_{moon} \) is the moon’s gravitational acceleration (which equals \(g/6\)).

Step 3: Calculate the work done to move the module to the specified height

The work done on an object is the product of the force applied and the distance over which it is applied. Therefore, using the formula \[ W = F \cdot d \], where \( W \) is work done, \( F \) is the force calculated in step 2 and \( d \) is the distance (50 miles, converted to meters since distances in the work formula must be in the same unit as in the force formula, hence convert 50 miles to meters considering 1 mile equal to 1609.34 meters).

Key Concepts

Gravitational Force

The gravitational force is a fundamental concept in physics that refers to the attractive force between two bodies with mass. This force is universal, meaning it acts between all objects with mass in the universe. When considering objects like a lunar module, it's essential to understand how gravitational force works. Gravitational force is determined by the masses of two bodies and the distance between them.

• The equation for gravitational force is given by Newton's law of universal gravitation: \[ F = G \frac{m_1 m_2}{r^2} \] where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the objects, and \( r \) is the distance between the centers of the two masses.
• On the Moon, gravitational force is weaker due to its smaller mass compared to the Earth, which significantly affects how objects like a lunar module behave when launched from its surface.
• This lower gravitational pull means that objects on the Moon weigh less, which is pivotal in calculations and operations involving space missions.

Lunar Module

A lunar module is a spacecraft designed specifically for travel between the lunar surface and orbit. When considering the work done in physics for a lunar module, it's crucial to factor in its design and purpose.

• It is constructed to endure the Moon's surface conditions, having lower gravitational forces and lack of atmosphere.
• Understanding a lunar module's movement involves knowing its mass and the gravitational forces acting upon it, especially when calculating the work done during propulsion.
• The mass of the lunar module remains constant, but its weight changes based on the gravitational pull of different celestial bodies like the Earth or the Moon.

Working with lunar modules also involves comprehending the relationship between mass, gravitational forces, and propulsion, particularly when planning missions to move modules from the Moon's surface to an orbital or escape velocity.

Mass and Weight

In physics, mass and weight are two distinct yet related concepts. It's important to differentiate between them to understand the dynamics of objects like the lunar module.

• Mass refers to the amount of matter in an object and it is constant, no matter where the object is located. Mass is measured in kilograms in the International System of Units (SI).
• Weight, on the other hand, is the force exerted by gravity on that mass. It changes depending on the gravitational pull of the body the object is on. It is calculated using the formula \( w = m \cdot g \), where \( w \) is weight, \( m \) is mass, and \( g \) is the gravitational acceleration.
• On the Moon, the lunar module will weigh less compared to its weight on Earth because the gravitational force is weaker there. This difference is crucial in performing calculations related to space missions.

Moon Gravity

Moon gravity is an intriguing subject because it is significantly weaker than Earth's gravity, being only about one-sixth (\( 1/6 \)of Earth's gravitational force.

• This reduced gravity is due to the Moon's smaller size and mass compared to Earth. On the Moon, objects fall slower, and they weigh much less than they do on Earth.
• For the lunar module, this means that propulsion requires calculating the new weight under moon gravity. This is a crucial step in ensuring the right amount of force is applied to achieve desired movements, such as launching 50 miles away from the Moon’s surface.
• Knowing the moon's gravity also helps in planning for fuel requirements and propulsion force needed for lunar missions, as it affects the work calculations involved in moving or lifting objects.

These aspects make moon gravity a vital consideration in space explorations and engineering of space vehicles.