Weight

Prabhav Agrawal Fall 2024

The Main Idea

In physics, weight describes the gravitational force exerted on a mass, typically relative to Earth or another celestial body. Weight can either be defined as a scalar—representing only the magnitude of the gravitational force—or as a vector, accounting for both magnitude and direction, as it always points toward the center of the gravitational field.

An object's weight is frequently misunderstood as its mass, but the two are fundamentally different. While mass is an intrinsic property of matter, representing the amount of substance within an object and remaining constant regardless of location, weight is a force that arises from the interaction between mass and a gravitational field. Weight depends not only on the object's mass but also on the gravitational strength of the body exerting the force.

For instance, on Earth, the weight of an object is determined by multiplying its mass by Earth's gravitational acceleration, approximately 9.8 m/s². On the Moon, where gravitational acceleration is only about 1.63 m/s², the same object will weigh significantly less. Similarly, in deep space, far from any major celestial bodies, an object could effectively become weightless, though its mass remains unchanged.

The concept of weight has profound implications across numerous scientific and practical applications. In space exploration, for example, weightlessness presents unique challenges for both human physiology and material design. On Earth, the understanding of weight is crucial in engineering disciplines, such as determining load capacities in construction, optimizing fuel consumption in transportation, and ensuring the safety and stability of machines.

A Mathematical Model

A mass m's weight near the surface of the Earth is represented by [math]\displaystyle{ {\vec{W} = \vec{F}_{g} = {m}\vec{g}} }[/math], where g is the gravitational acceleration of Earth, approximately [math]\displaystyle{ {{\lt 0,-9.8,0\gt } \frac{m}{{s}^{2}}} }[/math].

Scalar weight represents only the magnitude of the gravitational force, expressed as: [math]\displaystyle{ {\left\vert{\vec{W}}\right\vert = \left\vert{\vec{F}_{g}}\right\vert} }[/math], which simplifies to: [math]\displaystyle{ {\left\vert{\vec{W}}\right\vert = mg} }[/math].

Weight in Different Gravitational Environments In other gravitational environments, the weight of an object is determined using the local gravitational acceleration: [math]\displaystyle{ {\vec{W} = m\vec{g}{local}} }[/math], where [math]\displaystyle{ \vec{g}{local} }[/math] is specific to the celestial body in question. For example:

On the Moon, [math]\displaystyle{ \vec{g}_{local} ≈ 1.63 \frac{m}{{s}^{2}} }[/math].

On Mars, [math]\displaystyle{ \vec{g}_{local} ≈ 3.75 \frac{m}{{s}^{2}} }[/math].

On Jupiter, [math]\displaystyle{ \vec{g}_{local} ≈ 24.79 \frac{m}{{s}^{2}} }[/math].

The variation in weight is proportional to the ratio of the celestial body's gravitational acceleration to that of Earth.

Altitude and Weight Weight is not constant even on Earth. As altitude increases, the gravitational acceleration decreases due to the inverse-square law: [math]\displaystyle{ {g = G\frac{M}{R^2}} }[/math], where:

[math]\displaystyle{ G }[/math] is the universal gravitational constant, [math]\displaystyle{ 6.67430 \times 10^{-11} \frac{m^3}{kg \cdot s^2} }[/math], [math]\displaystyle{ M }[/math] is the Earth's mass, and [math]\displaystyle{ R }[/math] is the distance from the Earth's center. At higher altitudes, the value of [math]\displaystyle{ R }[/math] increases, causing [math]\displaystyle{ g }[/math] to decrease, which in turn reduces the weight.

Weight on the Earth's Surface For small variations in height (e.g., standing on a mountain), weight can be approximated using: [math]\displaystyle{ {\Delta W = W\left(1 - \frac{2h}{R}\right)} }[/math], where [math]\displaystyle{ h }[/math] is the height above sea level and [math]\displaystyle{ R }[/math] is the Earth's radius.

Examples of Weight Calculation Object on the Moon: A 10 kg object has a weight on the Moon given by: [math]\displaystyle{ {\vec{W}{Moon} = m\vec{g}{Moon} = 10 kg \times 1.63 \frac{m}{{s}^{2}} = 16.3 N} }[/math].

Object at High Altitude on Earth: For a 50 kg object at an altitude of 5,000 m above sea level (Earth’s radius [math]\displaystyle{ R = 6.371 \times 10^6 m }[/math]): [math]\displaystyle{ {g = G\frac{M}{(R + h)^2} \approx 9.77 \frac{m}{s^2}} }[/math]. The weight becomes: [math]\displaystyle{ {W = mg = 50 kg \times 9.77 \frac{m}{s^2} = 488.5 N} }[/math].

Weightlessness and Apparent Weight In free fall or in orbit, objects experience apparent weightlessness, as the only force acting on them is gravity. The apparent weight is effectively zero, even though the object still has mass and gravitational force acting on it. This phenomenon is crucial for understanding motion in space and in environments like the International Space Station.

By considering these factors, the concept of weight extends beyond Earth, providing insights into forces experienced across the universe.

A Computational Model

Below is a code snippet that calculates both scalar and vector weight (gravitational force) exerted upon a spherical object.

# Importing the required libraries from vpython import sphere, vec, color, mag python Copy code

# Initializing sphere object
ball = sphere(pos=vec(0,0,0), radius=0.02, color=color.yellow, make_trail=True)

# Defining constants
g = vec(0, -9.8, 0)  # Gravitational acceleration on Earth (m/s^2)
ball.m = 0.1          # Mass of the ball in kg
W = ball.m * g        # Weight of the ball on Earth

# Printing values
print("Scalar weight of the ball:", mag(W), "kg m/s^2 or N")
print("Force of gravity exerted on the ball:", W, "kg m/s^2 or N")

Code Extensions

The following code snippet calculates weight in other gravitational environments, such as on the Moon or Mars.

# Calculating weight on the Moon g_moon = vec(0, -1.62, 0) # Gravitational acceleration on the Moon (m/s^2) W_moon = ball.m * g_moon print("Scalar weight of the ball on the Moon:", mag(W_moon), "N") python Copy code

# Calculating weight on Mars
g_mars = vec(0, -3.75, 0)  # Gravitational acceleration on Mars (m/s^2)
W_mars = ball.m * g_mars
print("Scalar weight of the ball on Mars:", mag(W_mars), "N")

Practical Scenarios

Weight is crucial in assessing load capacities and measuring forces for various applications in engineering, science, and everyday use.

Engineering Applications

Structural Design: Engineers calculate weight for determining load-bearing capacities in buildings and bridges, considering gravitational forces.

Vehicle Design: Weight is critical in designing airplanes, rockets, and other vehicles for efficiency and stability.

Transportation: Understanding weight helps in determining load capacities for trucks and trains to ensure safe transport.

Scientific Research

Material Science: Weight-to-strength ratios are used to design lightweight or heavy-duty materials for specific applications.

Physics Experiments: Weight is measured in experiments to study gravitational forces and their effects on different objects.

Space Exploration

Spacecraft and Satellites: Engineers account for weight to optimize spacecraft performance and fuel efficiency in different gravitational environments.

Mars Rovers: Weight calculations are essential for ensuring proper function and performance on Mars, where gravity is weaker than on Earth.

Everyday Uses

Healthcare: Weight measurements help calculate medication dosages and monitor health.

Fitness: Weight tracking is used to assess fitness progress and body composition.

Example

Most problems involving weight calculation are simple; complex problems usually involve gravitational force in diverse contexts such as orbital mechanics or planetary exploration.

Simple

Determine the weight in Newtons of a 75-kilogram astronaut on the surface of the Moon, given the gravitational acceleration [math]\displaystyle{ {g}_{Moon} = 1.62\frac{m}{{s}^{2}} }[/math].

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = m{g}_{Moon} }[/math]

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = 75kg * 1.62\frac{m}{{s}^{2}} }[/math]

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = 121.5 N }[/math]

Advanced

Calculate the weight of a 10,000 kg satellite in low Earth orbit, where the effective gravitational acceleration is [math]\displaystyle{ {g}_{orbit} = 8.7\frac{m}{{s}^{2}} }[/math].

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = m{g}_{orbit} }[/math]

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = 10,000kg * 8.7\frac{m}{{s}^{2}} }[/math]

[math]\displaystyle{ \left\vert{\vec{W}}\right\vert = 87,000 N }[/math]

Connectedness

How is this topic connected to something that you are interested in?
1. Weight plays a pivotal role in areas like space exploration, where understanding weightlessness and its impact on materials is crucial. For example, designing materials for satellites or spacecraft must account for varying gravitational forces.
2. Gravitational force and weight are integral in the study of planetary science, helping scientists understand how celestial bodies interact and behave within a gravitational field.
3. Engineering new technologies, such as hyperloop transport systems, requires precise calculations of weight to ensure safety and efficiency in high-speed environments.
How is it connected to your major?
1. Materials science involves physics, chemistry, and engineering. Understanding weight helps in designing materials with specific properties for different applications, such as lightweight materials for aerospace engineering or heavy-duty materials for structural applications.
2. Gravitational force and weight influence material testing methods, such as tensile strength and durability tests, which are critical in assessing material performance under real-world conditions.
3. Research into nanomaterials often considers the weight-to-strength ratio, which is essential for applications in lightweight, high-performance products like medical implants or carbon-fiber composites.
Is there an interesting industrial application?
1. Besides weigh stations, modern robotics and drones use weight and gravitational force calculations to determine payload capacities and optimize performance.
2. Advanced industrial applications also use weight measurements in additive manufacturing to ensure precision and stability.
3. Agricultural industries employ weight-based sensors in automated sorting systems to grade and process crops efficiently.
4. Shipping and logistics industries rely heavily on weight calculations to optimize fuel usage, route planning, and load balancing, reducing costs and environmental impact.

History

Since weight is essentially the force of gravitation, refer to Gravitational Force for more about the history of the Law of Gravitation.

Historical advancements in understanding weight and gravity have contributed significantly to fields like astronomy, navigation, and modern physics.