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How do the gravitational force between masses and the electric force between charges depend on size and distance, and why are they so similar?

Describe Newton's law of gravitation and Coulomb's law, and use proportional reasoning to predict how the gravitational and electric forces change with mass, charge, and distance (MA STE Introductory Physics, HS-PS2-4).

A standard-level answer on Newton's law of gravitation and Coulomb's law for the Massachusetts High School Introductory Physics MCAS: how each force depends on size and distance, the inverse-square relationship, and how they compare under HS-PS2-4.

Generated by Claude Opus 4.812 min answer

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  1. What this topic is asking
  2. Newton's law of gravitation
  3. Coulomb's law
  4. The inverse-square relationship
  5. Reference-sheet note
  6. Try this

What this topic is asking

Two of the fundamental forces of nature, gravity and the electric force, follow the same mathematical pattern, and the Massachusetts Introductory Physics standard HS-PS2-4 asks you to recognize it. You must describe Newton's law of gravitation and Coulomb's law, and use proportional reasoning to predict how each force changes when the masses, charges, or distance change. The MCAS does not ask you to plug into the full formulas (they are not on the reference sheet); it tests the proportional relationships, especially the inverse-square dependence on distance.

Newton's law of gravitation

In words, the gravitational force grows if either mass grows, and it weakens rapidly as the objects move apart. Gravity is by far the weakest of the fundamental forces, so the attraction between two everyday objects is far too small to feel. It becomes important only when at least one of the masses is huge, like a planet or a star. This is why you feel the Earth's gravity (your weight) but not the gravitational pull of the person next to you, even though that pull genuinely exists.

Coulomb's law

Coulomb's law has the same mathematical shape as the law of gravitation: a product of the two "sources" (charges, instead of masses) divided by the distance squared. The big differences are that the electric force can attract or repel (gravity only attracts), and the electric force is enormously stronger than gravity for the same separation. The matching form is exactly what HS-PS2-4 highlights: two very different forces obey the same kind of rule.

The inverse-square relationship

Proportional reasoning is the way the MCAS tests these laws, because the full formulas are not given. Work through the change one factor at a time:

  • Change a mass or a charge by a factor: the force changes by the same factor (direct proportion).
  • Change the distance by a factor: the force changes by one over that factor squared (inverse square).

Reference-sheet note

The full equations for the law of gravitation and Coulomb's law are not on the Introductory Physics reference sheet, so the MCAS tests them qualitatively and with proportional reasoning rather than by direct substitution. What you need to recall is the form of each law (proportional to the sources, inverse square in distance), the directions (gravity attracts; like charges repel, opposite charges attract), and the relative strengths (gravity weak, electric force strong).

Try this

Q1. State what happens to the gravitational force between two objects if the distance between them is tripled. [1]

  • Cue. The force becomes one ninth (32=93^2 = 9, inverse square).

Q2. Two charges attract each other. State whether they have like or opposite signs. [1]

  • Cue. Opposite signs (opposite charges attract; like charges repel).

Exam-style practice questions

Practice questions written in the style of MA DESE exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

MA Physics MCAS (style)3 marksTwo objects attract each other gravitationally. (a) State what happens to the gravitational force if the mass of one object is doubled. (b) State what happens to the force if the distance between them is doubled. (c) Explain why the gravitational force between everyday objects is normally too small to notice.
Show worked answer →

A 3-point item on proportional reasoning with the law of gravitation.

(a) 1 point: the force doubles, because gravitational force is proportional to the mass of each object.
(b) 1 point: the force becomes one quarter, because it depends on the inverse square of the distance (22=42^2 = 4, so the force is divided by 44).
(c) 1 point: gravity is an extremely weak force, so the force between two ordinary masses is tiny; it only becomes noticeable when at least one mass is enormous, like a planet. Markers reward the inverse-square reasoning in (b) and the idea that gravity is weak in (c).

MA Physics MCAS (style)2 marksTwo charged spheres repel each other. (a) State whether the charges have the same sign or opposite signs. (b) State what happens to the electric force between them if the distance is halved.
Show worked answer →

A 2-point item on Coulomb's law and the inverse-square relationship.

(a) 1 point: the charges have the same sign (like charges repel; opposite charges attract).
(b) 1 point: halving the distance multiplies the force by four, because the electric force depends on the inverse square of the distance, and halving the distance gives (12)2=14\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4} in the denominator, so the force is multiplied by 44. Markers reward the factor of four and the inverse-square idea.

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