Two point masses, each of mass 125\text{ kg}, are fixed at positions (-3.0\text{ m}, 0) an… — JAMB UTME

Two point masses, each of mass $125\text{ kg}$ , are fixed at positions $(-3.0\text{ m}, 0)$ and $(3.0\text{ m}, 0)$ in the $xy$ -plane. What is the magnitude of the net gravitational field strength at a point $P$ located at $(0, 4.0\text{ m})$ , in terms of the universal gravitational constant $G$ ?

$8.0G\text{ N/kg}$ Cevap
B
$10.0G\text{ N/kg}$
C
$6.0G\text{ N/kg}$
D
$4.0G\text{ N/kg}$

Cevap

The magnitude of the net gravitational field strength at point

P

8.0G\text{ N/kg}

The distance from each mass to the target point is 5.0 m, yielding an individual field magnitude of 5.0G N/kg. Resolving vectors shows that horizontal components cancel out while vertical components add to give 2 × (5.0G × 4/5) = 8.0G N/kg.

Adım Adım Çözüm

Calculate the distance

r

from each mass to point

P(0, 4.0\text{ m})

r = \sqrt{(-3.0 - 0)^2 + (0 - 4.0)^2} = \sqrt{9 + 16} = 5.0\text{ m}

The distance formula in two dimensions gives the hypotenuse of the right triangle formed by the coordinates.

Determine the magnitude of the gravitational field

E

created by each individual mass at point

P

E = \frac{G M}{r^2} = \frac{G \times 125}{5.0^2} = \frac{125G}{25} = 5.0G\text{ N/kg}

Newton's law of universal gravitation defines field strength as

E = \frac{GM}{r^2}

Resolve the field vectors into horizontal and vertical components.

By symmetry, the horizontal components

E_x = E \sin\theta

are equal in magnitude and opposite in direction, so

E_{x,\text{net}} = 0

. The vertical components

E_y = E \cos\theta

point downwards towards the origin.

Gravitational field is a vector quantity, so opposite components cancel while aligned components add together.

Calculate the total vertical component of the net gravitational field.

\cos\theta = \frac{4.0}{5.0} = 0.8

. Thus,

E_{\text{net}} = 2 \times E_y = 2 \times (5.0G \times 0.8) = 8.0G\text{ N/kg}

Summing the vertical contributions from both identical masses gives the net magnitude.

Anahtar Kavram

Vector Addition of Gravitational Field Strengths

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