how-to · 6 min read
Gravitation, heat and lenses: understanding three core physics formulas
Gravitation, heat transfer and lenses look like separate topics, but each is a compact formula with a small set of symbols, a sign convention and a set of assumptions. Learning the three together is a useful exercise because the same questions keep returning: what does the sign mean, which units must match, and when does the model stop describing reality?
This guide works through each formula with a concrete example, then explains the assumptions well enough that a calculated answer can be judged rather than simply trusted.
Gravity is an inverse-square law
Newton's law of gravitation gives the attraction between two masses: F = G·m₁·m₂/r², with G = 6.67430 × 10⁻¹¹ N·m²/kg². The force grows with either mass but falls with the square of the distance between their centers, so moving two bodies twice as far apart leaves only one quarter of the force. At three times the distance it is one ninth.
The inverse square comes from geometry: the influence of a mass spreads over a sphere whose area grows with r², so its strength per unit area falls as 1/r². The same pattern appears in light intensity and sound spreading from a source.
The Earth and Sun are a useful worked example. Using the Earth's mass (about 5.97 × 10²⁴ kg), the Sun's mass (about 1.99 × 10³⁰ kg) and an average separation of about 1.5 × 10¹¹ m gives a force near 3.5 × 10²² N. The same force pulls each body, but the accelerations differ: a = F/m, so the Earth's acceleration is tiny compared with the Sun's response to it.
A uniform sphere behaves like a point mass located at its center for anything outside it, which is why the point-mass form works for planets and stars. Inside a body, or between irregular shapes very close together, the simple formula becomes an approximation.
Heat transfer and the sign of q
Specific heat capacity describes how much energy one kilogram of a material needs for a one-kelvin temperature change. The relation is q = m·c·ΔT, with mass in kilograms, c in J/(kg·K) and the temperature change in kelvin. A temperature interval of one degree Celsius is the same size as an interval of one kelvin, so either scale can be used for ΔT.
Water has a high specific heat, about 4,184 J/(kg·K), which is why it heats slowly and stores a lot of energy for a modest temperature rise. Metals are much lower: aluminum is around 900 J/(kg·K), copper around 385 J/(kg·K) and iron around 449 J/(kg·K).
As a worked example, heating 0.5 kg of water from 20 °C to 80 °C means ΔT = 60 K, so q = 0.5 × 4,184 × 60, which is about 125,520 J, or roughly 1.26 × 10⁵ J. The temperature rises, so q is positive and heat is gained.
The sign is a direction, not an error. If the same water cools from 80 °C to 20 °C, ΔT is −60 K and q is negative: the water releases the same amount of energy to its surroundings. During a phase change such as melting or boiling, the temperature stays constant while energy breaks intermolecular bonds, and the q = mcΔT relation does not apply within that transition.
The thin-lens equation and what the signs mean
For a thin lens, 1/f = 1/dₒ + 1/dᵢ, with the focal length f and the object and image distances dₒ and dᵢ. The magnification is m = −dᵢ/dₒ, so a negative m indicates an inverted image and |m| tells you whether it is enlarged or reduced.
The Cartesian sign convention keeps the algebra consistent: light travels left to right, dₒ is positive for a real object in front of the lens, f is positive for a converging lens and negative for a diverging lens, and dᵢ is positive for a real image on the far side and negative for a virtual image on the object side.
With f = 10 cm and an object at dₒ = 30 cm, the equation gives 1/dᵢ = 1/10 − 1/30 = 1/15, so dᵢ = 15 cm. The image is real, inverted (m = −0.5) and half the object's size.
Move the object inside the focal point, to dₒ = 5 cm, and 1/dᵢ = 1/10 − 1/5 = −1/10, so dᵢ = −10 cm. The negative image distance means a virtual image on the same side as the object, with m = +2: upright, enlarged and impossible to project on a screen. That is exactly how a magnifying glass works.
- Positive focal length: converging lens; negative focal length: diverging lens.
- Positive image distance: real image on the far side; negative: virtual image on the object side.
- Negative magnification: inverted image; positive: upright image.
- Object at the focal point: rays leave parallel and no finite image forms.
Where the models stop being exact
All three formulas are idealizations. The gravitational formula assumes point masses or uniform spheres, so it ignores the shape of irregular asteroids, the bulge of a rotating planet and the pull of a third body. It is an excellent approximation for orbits and surface gravity, and an incomplete one for a multi-body mission design.
The heat equation assumes a single phase and no heat exchange with the surroundings. Real calorimetry has container heat capacity, heat lost to the air and temperature-dependent specific heat, which is why measured values usually differ from a simple calculation. The larger the temperature range, the more these effects matter.
The thin-lens equation assumes a thin lens and rays close to the optical axis. Thick lenses, wide apertures and off-axis rays produce spherical and chromatic aberrations that the paraxial model cannot capture, and multi-element lenses need ray tracing or design software.
None of this makes the formulas wrong for their intended job. They are compact models that explain the dominant behavior, give a first estimate and show which variable matters most. Use them to understand and check, then measure or simulate when precision matters.