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PhysicsMadrid 2026Question 2.BFree lesson

Gravitational field of two point masses

Where the field vanishes and how to add two fields that are not parallel: why it is solved this way

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Problem statement

Question 2.B.- Two point masses, $m_1 = 2$ kg and $m_2 = 4$ kg, are placed at the origin of coordinates and at the point $(4,\,0)$ m of the $xy$ plane, respectively. Calculate:

a) (1.5 marks) The point on the $x$ axis between the two masses at which the gravitational field created by both masses is zero.

b) (1 mark) The total gravitational field due to both masses at the point $(0,\,2)$ m.

Data: Universal Gravitation Constant, $G = 6.67\cdot10^{-11}\ \text{N m}^2\ \text{kg}^{-2}$.

Source: Madrid university entrance exam (PAU, Spain) · 2025-2026 · Ordinary session · Physics · Question 2.B. Official exam set by the public universities of the Community of Madrid, to whom it belongs; translated from Spanish and reproduced with its source in order to explain it for teaching purposes.

What this lesson answers

  1. What is the gravitational field, and what can we anticipate without calculating?
  2. a) Why can the field only vanish between the two masses?
  3. a) Where do the magnitudes become equal, and why is it convenient to take the square root?
  4. a) Another route: the quadratic equation and the extra solution
  5. b) Why must we add vectors, not magnitudes, at point P?
  6. b) Where do the components of the large mass's field come from?
  7. b) Another route: the vector formula, without angles
  8. The two answers and how to check them
  9. Common mistakes to avoid
  10. Exam tips
  11. If you want top marks

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