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If $0 < st < 1$, which of the following could be true?
We need $0 < st < 1$, meaning $st$ is a small positive number. This requires $s$ and $t$ to have the same sign.
- Option A: $s < -1, t > 0$ → $st < 0$. Not possible.
- Option B: $s < -1, t < -1$ → both negative, so $st > 0$. For example $s = -2, t = -0.3$: $st = 0.6$, which satisfies $0 < st < 1$. ✓
- Option C: $s > -1, t < -1$ → could give $st$ negative or positive depending on sign of $s$. If $s = 0.5, t = -2$: $st = -1 < 0$. Not valid for all cases.
- Option D: $s > 1, t < -1$ → $st < -1 < 0$. Not possible.
Option B is the one that can satisfy the condition.
Planck's constant ($h$), speed of light in vacuum ($c$), and Newton's gravitational constant ($G$) are taken as fundamental constants. Which combination of these has the dimension of length?
This is the Planck length $\ell_P = \sqrt{\dfrac{\hbar G}{c^3}}$ (with $h$ in place of $\hbar$ up to a constant).
Dimensions: $[h] = ML^2T^{-1}$, $[G] = M^{-1}L^3T^{-2}$, $[c] = LT^{-1}$
$\left[\dfrac{hG}{c^3}\right] = \dfrac{ML^2T^{-1} \cdot M^{-1}L^3T^{-2}}{L^3T^{-3}} = \dfrac{L^5T^{-3}}{L^3T^{-3}} = L^2$
$\Rightarrow \sqrt{\dfrac{hG}{c^3}} \sim [L]$ ✓
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