Weight

Prabhav Agrawal Fall 2024

The Main Idea

In physics, weight describes the Gravitational Force upon a mass, usually relative to Earth or a planet. Depending on the textbook, weight may be defined as a scalar - the magnitude of the gravitational force on an object - or a vector equal to gravitational force.

An object's weight is commonly confused with its mass, but instead, it is a force that depends on another body of matter, while mass is an intrinsic property that measures the amount of matter within an object.

Weight varies depending on the strength of the gravitational field in which an object is located. For example, the same object will weigh less on the Moon than on Earth due to the Moon's weaker gravitational pull.

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, [math]\displaystyle{ {{\lt 0,-9.8,0\gt } \frac{m}{{s}^{2}}} }[/math].

Scalar weight would simply be the magnitude of the gravitational force, [math]\displaystyle{ {\left\vert{\vec{W}}\right\vert = \left\vert{\vec{F}_{g}}\right\vert} }[/math], and it can be simplified to [math]\displaystyle{ {\left\vert{\vec{W}}\right\vert = mg} }[/math].

In other gravitational environments, the weight can be determined using the local gravitational acceleration: [math]\displaystyle{ {\vec{W} = m\vec{g}{local}} }[/math], where [math]\displaystyle{ \vec{g}{local} }[/math] varies depending on the planet, moon, or other celestial body.

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")

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Practical Scenarios

Weight can be used to assess load capacities in engineering or to measure forces in scientific experiments. This knowledge is crucial in designing structures, vehicles, and materials for various gravitational environments.

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

1. 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.
2. 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.
3. 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.

See also

Vectors

Mass

Gravitational Force

Gravitational Potential Energy

Weightlessness

Further reading

Chabay & Sherwood: Matters and Interactions -- Modern Mechanics Volume 1, 4th Edition Serway & Jewett: Physics for Scientists and Engineers

External links

Physics Classroom lessons and notes NASA resources on weight and gravity

References

1. "Weight." Wikipedia. Wikimedia Foundation. Web. 1 Dec. 2015. https://en.wikipedia.org/wiki/Weight#Gravitational_definition.
2. "Types of Forces." Types of Forces. Physics Classroom. Web. 1 Dec. 2015. http://www.physicsclassroom.com/class/newtlaws/Lesson-2/Types-of-Forces.
3. "The Value of G." The Value of G. Physics Classroom. Web. 1 Dec. 2015. http://www.physicsclassroom.com/class/circles/Lesson-3/The-Value-of-g.
4. "How do truck weigh stations work?" 01 May 2001. HowStuffWorks.com. Web. 1 Dec. 2015.http://science.howstuffworks.com/engineering/civil/question626.htm