Interactive physics simulator
Weight: W = mg
Investigate gravity's pulling force on matter. Test dynamic spring scale extensions under planetary gravities, ride an accelerating elevator to track apparent weight fluctuations, and compare mass conservation side-by-side on Earth, Moon, and Jupiter scales.
Weight Laboratory
Select celestial gravity, hang items on spring scales, or accelerate the elevator to analyze forces and spring compressions.
Live Telemetry
- Mass (m)
- 10.0 kg
- Gravity (g)
- 9.81 m/s²
- True Weight (W)
- 98.1 N
- Spring Constant (k)
- 250 N/m
- Spring Stretch (x)
- 39.2 cm
- Scale Reading
- 98.1 N
What is Weight? (W = mg)
In physics, weight ($W$) is the gravitational force exerted on an object by a massive celestial body (like Earth or the Moon). It is directly proportional to both the object's **mass** ($m$) and the local **acceleration due to gravity** ($g$):
Where:
- W: Weight, measured in Newtons (N). Since it is a force, it is a vector quantity pointing straight down towards the planet's center of mass.
- m: Mass, measured in kilograms (kg). It is a scalar quantity measuring the quantity of matter in the object, which is completely invariant.
- g: Gravitational acceleration field, measured in m/s² or N/kg (approximately 9.81 m/s² on Earth's surface).
Mass vs Weight
Comparing physical property vs gravitational force:
- Mass: Invariant. A 10 kg block remains 10 kg in space, on the Moon, or on Jupiter. Measured with standard pan balances.
- Weight: Variable. The same 10 kg block weighs 98 N on Earth, but only 16 N on the Moon, and 248 N on Jupiter.
- Measurement tool: Spring scales measure weight by reading spring deflection ($F = kx$), which changes depending on local gravity.
Elevator Apparent Weight
Feeling heavier or lighter during acceleration:
- Scale Reaction: Weighing scales register normal force ($F_n$), which represents your **apparent weight**.
- Upward Acceleration: scale pushes extra hard ⇒ $F_n = m(g + a)$. You feel heavier than normal.
- Downward Acceleration: scale drops away ⇒ $F_n = m(g - a)$. You feel lighter.
- Snapping Cable: In free fall ($a = -g$), the normal force drops to 0. You float, experiencing complete weightlessness.
Spring Scale Extension
Hooke's Law and force equilibriums:
- Restoring Force: Stretching a spring scale creates a force $F_s = -kx$ opposing gravity.
- Static Equilibrium: When the hung mass stops bouncing, spring force balances weight: $kx = mg$.
- Extension length: The stretch distance is directly proportional to weight: $x = mg / k$. Doubling the mass doubles the stretch.
Inertial vs Gravitational Mass
Two distinct concepts with identical values:
- Inertial Mass: Resistance to being accelerated by a force ($m_i = F / a$). Tested in deep space centrifuge systems.
- Gravitational Mass: Attracting sensitivity to gravity fields ($m_g = W / g$). Tested using spring scale weights.
- Equivalence Principle: Einstein showed $m_i \equiv m_g$. A rocket accelerating at 9.81 m/s² generates a force identical to Earth's weight.
Solved Examples
An astronaut with a spacesuit has a total mass of 120 kg. Calculate their weight on Earth (g = 9.81 m/s²) and on the Moon (g = 1.62 m/s²). How does their mass change?
- Identify the mass: m = 120 kg. Mass is the quantity of matter and remains exactly constant (120 kg) on both Earth and the Moon.
- Use the weight formula: W = mg.
- On Earth: W_earth = 120 kg * 9.81 m/s² = 1177.2 Newtons.
- On the Moon: W_moon = 120 kg * 1.62 m/s² = 194.4 Newtons.
- Verify: The astronaut’s mass remains 120 kg, but they weigh about 6 times less on the Moon due to the weaker local gravitational field.
Answer: Mass = 120 kg on both; Weight on Earth = 1177.2 N, Weight on Moon = 194.4 N
A 70 kg student stands on a weighing scale inside an elevator. Calculate the apparent weight read by the scale if the elevator is: (a) moving at a constant speed of 3.0 m/s upward, (b) accelerating upward at 2.5 m/s², and (c) accelerating downward at 2.0 m/s². (g = 9.81 m/s²)
- Identify initial values: mass m = 70 kg, gravity g = 9.81 m/s².
- (a) Constant speed means acceleration a = 0. Apparent weight equals true weight: W_app = m * g = 70 * 9.81 = 686.7 Newtons.
- (b) Upward acceleration (a = +2.5 m/s²): The scale must exert extra force to accelerate the student. W_app = m(g + a) = 70 * (9.81 + 2.5) = 70 * 12.31 = 861.7 Newtons (feels heavier).
- (c) Downward acceleration (a = -2.0 m/s²): The scale supports less weight. W_app = m(g - a) = 70 * (9.81 - 2.0) = 70 * 7.81 = 546.7 Newtons (feels lighter).
Answer: (a) Apparent Weight = 686.7 N, (b) Apparent Weight = 861.7 N, (c) Apparent Weight = 546.7 N
A spring scale has a spring constant of k = 400 N/m. If a student hangs a custom physics mass on the scale in a vacuum and it stretches the spring by exactly 12.2 cm, calculate the weight and the mass of the object. (g = 9.81 m/s²)
- Convert stretch distance to SI units: x = 12.2 cm = 0.122 meters.
- Apply Hooke's Law for a spring in static equilibrium (gravitational force equals spring restoring force): F_spring = k * x.
- Calculate the weight (force): W = k * x = 400 N/m * 0.122 m = 48.8 Newtons.
- Apply the weight formula to find mass: W = m * g ⇒ m = W / g.
- Substitute values: m = 48.8 N / 9.81 m/s² ≈ 4.97 kilograms.
Answer: Weight = 48.8 N, Mass ≈ 4.97 kg
Common Misconceptions
- "Mass changes on the Moon": False. Mass is an invariant amount of matter. Only the force of gravity (weight) changes.
- "Zero gravity in orbit causes floating": False. Gravity on the International Space Station is about 90% of Earth\'s surface value. Astronauts float because they are in permanent free fall together with the station.
- "Weight is measured in kilograms": False. In daily life, we use kg for weight, but in physics, weight is a force and must be in Newtons. Kilogram is strictly for mass.
Practice Questions
1. Explain why a person’s weight is different on the equator versus the poles of the Earth, and whether their mass changes.
A person's mass represents the amount of matter in their body, which remains absolutely constant regardless of location. However, their weight changes due to two factors:
1. Earth's Shape: Earth is an oblate spheroid. The poles are closer to Earth's center of mass than the equator, making gravity slightly stronger at the poles.
2. Centrifugal Force: The Earth's rotation creates a centrifugal force that acts outwards, opposing gravity, with a maximum effect at the equator and zero at the poles.
Consequently, acceleration due to gravity (g) is about 9.78 m/s² at the equator and 9.83 m/s² at the poles, making a person weigh slightly more at the poles.
2. Describe the physical operation of a spring scale versus a double-pan balance. Which one measures mass, which measures weight, and why?
1. Spring Scale: Measures weight. It operates by stretching a spring according to Hooke's Law (F = kx). Since the stretching force is gravity pulling the mass down, the scale directly measures the local gravitational force. If taken to the Moon, it will read 1/6th of the Earth value.
2. Double-Pan Balance: Measures mass. It compares the gravitational pull on an unknown object in one pan against standard reference masses in the other pan. Since local gravity (g) affects both pans equally (canceling out in the torque equilibrium equation), it yields the exact same mass reading regardless of the local gravity field, making it ideal for measuring invariant mass.
3. Derive the apparent weight equation for a scale inside an elevator that is in free fall. What does a person stand on the scale experience?
An object in an accelerating elevator experiences two vertical forces: gravity downward (mg) and normal support force upward (F_n). Applying Newton's second law (upward positive): F_n - mg = ma ⇒ F_n = m(g + a).
If the elevator cable snaps, it enters a state of free fall, accelerating downward at the rate of gravity, so a = -g.
Substituting this in gives: F_n = m(g + (-g)) = m(0) = 0.
Since the scale registers the normal force (F_n), the scale reads exactly zero. The person floats and experiences a state of complete weightlessness.
4. Explain the conceptually profound distinction between inertial mass and gravitational mass, and how the Equivalence Principle links them.
1. Inertial Mass: Measures an object's resistance to acceleration when any mechanical force is applied (governed by F = ma). It is a property of mass that has nothing to do with gravity.
2. Gravitational Mass: Measures an object's sensitivity to gravitational fields—both how much gravitational force it exerts and how much it experiences (governed by F = G*m1*m2 / r²).
For centuries, physicists wondered why the mass that resists acceleration is the exact same mass that responds to gravity. Albert Einstein formulated the Equivalence Principle, stating that inertial and gravitational mass are completely identical, meaning acceleration and gravity are physically indistinguishable, which became the cornerstone of General Relativity.
FAQ
Frequently Asked Questions
What is the difference between mass and weight?
Mass is the amount of matter in an object and remains constant everywhere in the universe. Weight is the gravitational force acting on that mass and varies depending on the local gravitational strength (g).
What is the formula to calculate weight?
Weight is calculated using the formula W = mg, where W is the weight in Newtons (N), m is the mass in kilograms (kg), and g is the acceleration due to gravity (approximately 9.81 m/s² on Earth).
What is the SI unit of mass and weight?
The SI unit of mass is the kilogram (kg). Since weight is a force, its SI unit is the Newton (N).
Why does weight change on different planets?
Weight depends on the local gravitational acceleration constant (g). Since different planets have different masses and radii, their gravity varies (e.g., g is 1.62 m/s² on the Moon and 24.79 m/s² on Jupiter), causing an object’s weight to change accordingly.
How do we measure mass and weight?
Mass is typically measured using a double-pan balance by comparing it with known standard masses. Weight is measured using a spring scale (spring balance), which measures the force of gravity stretching or compressing a spring.
What is apparent weight in an elevator?
Apparent weight is the support force (normal force) exerted by a scale on an object. In an elevator accelerating upwards, the apparent weight increases (W = m(g + a)), making you feel heavier. In an elevator accelerating downwards, it decreases (W = m(g - a)), making you feel lighter.
Can weight be zero?
Yes. An object experiences weightlessness (apparent weight is zero) when in a state of free fall, such as in orbit, because there is no normal force supporting it. True weight is only zero when the gravitational field is zero, such as in deep space.
Is weight a scalar or vector quantity?
Weight is a vector quantity because it represents a force, which has both a magnitude and a direction (pointing vertically downwards towards the center of gravity of the host body). Mass is a scalar quantity.
What is the difference between gravitational mass and inertial mass?
Inertial mass measures an object’s resistance to acceleration when a force is applied (F = ma). Gravitational mass measures the strength of its gravitational attraction to other masses. Expermentally, these two masses are exactly equal (the equivalence principle).
How do you convert mass to weight on Earth?
On Earth, you multiply the mass in kilograms by approximately 9.81. For example, a 10 kg mass has a weight of 10 * 9.81 = 98.1 Newtons.
Does air buoyancy affect weight measurements?
Yes, slightly. When weighing an object in air, the buoyant force of the displaced air pushes upward, making the scale read slightly less than the object’s true weight in a vacuum. This effect is usually negligible for dense objects.