To find the minimum force $F$, let’s analyze the given equation:

$F = \frac{mg\mu}{\sin\theta + \mu \cos\theta}$

Given:

• $\mu = 0.54$
• Range for $\theta$: $0 \leq \theta \leq \frac{\pi}{2}$

Steps to find the minimum force:

1. Behavior of the denominator: The denominator $(\sin\theta + \mu \cos\theta)$ determines the minimum force. The force $F$ is inversely proportional to the denominator.

2. Maximizing the denominator: $\text{Maximize } \sin\theta + \mu \cos\theta$ Differentiate with respect to $\theta$ and find the critical points: $\frac{d}{d\theta} (\sin\theta + \mu \cos\theta) = \cos\theta - \mu \sin\theta$ Set the derivative equal to zero: $\cos\theta = \mu \sin\theta \quad \Rightarrow \quad \tan\theta = \frac{1}{\mu}$ Substitute $\mu = 0.54$: $\tan\theta = \frac{1}{0.54} \quad \Rightarrow \quad \theta = \arctan\left(\frac{1}{0.54}\right) \approx 61.3^\circ$

3. Evaluate the denominator at $\theta = 61.3^\circ$, $\theta = 0$, and $\theta = \frac{\pi}{2}$:

   • At $\theta = 0$: $\sin 0 + 0.54 \cos 0 = 0 + 0.54 = 0.54$
   • At $\theta = \frac{\pi}{2}$: $\sin \frac{\pi}{2} + 0.54 \cos \frac{\pi}{2} = 1 + 0 = 1$
   • At $\theta = 61.3^\circ$: Use $\sin 61.3^\circ \approx 0.874$ and $\cos 61.3^\circ \approx 0.485$: $\sin 61.3^\circ + 0.54 \cos 61.3^\circ = 0.874 + 0.54(0.485) \approx 1.135$

   The maximum denominator occurs at $\theta \approx 61.3^\circ$.

4. Calculate minimum force: Substitute the maximum denominator into the equation for $F$: $F_{\text{min}} = \frac{mg\mu}{\text{Max}(\sin\theta + \mu \cos\theta)}$ With $\sin 61.3^\circ + 0.54 \cos 61.3^\circ \approx 1.135$: $F_{\text{min}} = \frac{mg(0.54)}{1.135} \approx 0.475 \, mg$

Final Answer:

$\boxed{F_{\text{min}} \approx 0.475 \, mg \, \text{Newtons}}$

Would you like me to expand further on the derivation or provide additional examples?

Related Questions:

1. What happens to the force $F$ if $\mu$ decreases to 0.3?
2. Why is the minimum force not at $\theta = 0$ or $\theta = \frac{\pi}{2}$?
3. Can this result be applied if $\mu$ is larger than 1? Why or why not?
4. How does gravity $g$ impact the minimum and maximum forces?
5. What if the angle of pulling exceeds $\pi/2$? Would the formula still work?

Tip:

For any optimization problem involving trigonometric functions, analyze the behavior of $\sin\theta$ and $\cos\theta$ carefully over the given range.