Two point masses of 4.0\text{ kg} and 9.0\text{ kg} are separated by a distance of 5.0\tex… — JAMB UTME

Two point masses of $4.0\text{ kg}$ and $9.0\text{ kg}$ are separated by a distance of $5.0\text{ m}$ in free space. At what distance from the $4.0\text{ kg}$ mass along the line joining them is the net gravitational field strength equal to zero?

$2.0\text{ m}$ Answer
B
$3.0\text{ m}$
C
$2.5\text{ m}$
D
$1.0\text{ m}$

Answer

The distance from the

4.0\text{ kg}

mass where the net gravitational field strength is zero is

2.0\text{ m}

At the point where the net gravitational field strength is zero, the gravitational field intensity produced by the

4.0\text{ kg}

mass must equal the intensity produced by the

9.0\text{ kg}

mass in magnitude. Equating

\frac{G (4.0)}{x^2} = \frac{G (9.0)}{(5.0 - x)^2}

and taking square roots yields

\frac{2}{x} = \frac{3}{5 - x}

. Cross-multiplying gives

10 - 2x = 3x

, which yields

x = 2.0\text{ m}

from the

4.0\text{ kg}

mass.

Step-by-Step Solution

Set up the condition for zero net gravitational field strength.

E_1 = E_2 \implies \frac{G m_1}{x^2} = \frac{G m_2}{(d - x)^2}

At the neutral point, the opposing gravitational field vectors due to both masses are equal in magnitude.

Substitute given values

m_1 = 4.0\text{ kg}

m_2 = 9.0\text{ kg}

, and total distance

d = 5.0\text{ m}

\frac{4.0}{x^2} = \frac{9.0}{(5.0 - x)^2}

Simplifying by canceling

G

from both sides of the equation.

Take the square root of both sides and solve for

x

\frac{2.0}{x} = \frac{3.0}{5.0 - x} \implies 2.0(5.0 - x) = 3.0x \implies 10.0 - 2.0x = 3.0x \implies 5.0x = 10.0 \implies x = 2.0\text{ m}

Taking the square root simplifies the quadratic relationship into a linear ratio.

Key Concept

Gravitational Field Strength Neutral Point

Estimated Time:1m 0s

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