In a gravitational field, the gravitational potential is given by $V=-\frac{K}{x}$ J/kg. The gravitational fie

Question

In a gravitational field, the gravitational potential is given by $V=-\frac{K}{x}$ J/kg. The gravitational field intensity at the point $(2,0,3)$ m is:

Accepted Answer

1. **Relationship between Field and Potential:** The gravitational field intensity ($I$ or $E$) is the negative gradient of the gravitational potential ($V$). In one dimension (since $V$ depends only on $x$), the relationship is given by: 
   $$ I = -\frac{dV}{dx} $$
   (Reference: Concept of conservative forces, $F = -dV/dx$, applied to unit mass ).
2. **Differentiation:** Given $V = -Kx^{-1}$. Differentiating with respect to $x$:
   $$ \frac{dV}{dx} = \frac{d}{dx}(-Kx^{-1}) = -K(-1)x^{-2} = \frac{K}{x^2} $$
3. **Calculation:** Substitute the derivative back into the field equation:
   $$ I = -\left( \frac{K}{x^2} \right) = -\frac{K}{x^2} $$
4. **Substitution:** At the point $(2, 0, 3)$, the value of $x$ is $2$. Substitute $x=2$:
   $$ I = -\frac{K}{2^2} = -\frac{K}{4} $$
   The gravitational field intensity is $-\frac{K}{4}$ N/kg (or J/kg/m).

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