Problem 58 - Essential Calculus Early Transcendentals

Question

An object with weight \(W\) is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle \(\theta\) with the plane, then the magnitude of the force is $$F=\frac{\mu W}{\mu \sin \theta+\cos \theta}$$ where \(\mu\) is a positive constant called the coefficient of friction and where 0\(\leqslant \theta \leqslant \pi / 2 .\) Show that \(F\) is minimized when \(\tan \theta=\mu\)

Step-by-Step Solution

Answer

The force \(F\) is minimized when \(\tan \theta = \mu\).

Step 1: Understanding the Problem

We need to show that the force \(F = \frac{\mu W}{\mu \sin \theta + \cos \theta}\), is minimized when \(\tan \theta = \mu\). Our goal is to find the condition for \(\theta\) that minimizes \(F\).

Step 2: Expressing the Function

The function to minimize is \(F(\theta) = \frac{\mu W}{\mu \sin \theta + \cos \theta}\). We'll focus on minimizing the denominator, \(D(\theta) = \mu \sin \theta + \cos \theta\), since \(W\) is constant.

Step 3: Finding the Derivative of the Denominator

Differentiate the denominator with respect to \(\theta\):\[ D'(\theta) = \mu \cos \theta - \sin \theta \]

Step 4: Setting the Derivative to Zero for Critical Points

Solve \(D'(\theta) = 0\):\[ \mu \cos \theta = \sin \theta \]This gives us the ratio \(\tan \theta = \mu\).

Step 5: Second Derivative Test for Minimization

Find the second derivative of \(D(\theta)\):\[ D''(\theta) = -\mu \sin \theta - \cos \theta \]At \(\tan \theta = \mu\), check the sign of \(D''(\theta)\). Since \(D''(\theta)\) results in negative, \(D(\theta)\) is minimized, thus minimizing \(F(\theta)\).

Step 6: Conclusion

Thus, the force \(F\) is minimized when \(\tan \theta = \mu\). This occurs when the angle \(\theta\) makes \(F(\theta)\) as low as possible.

Key Concepts

Coefficient of Friction

The coefficient of friction, denoted as \( \mu \), is a crucial factor in understanding the resistance encountered when two surfaces slide against each other. It is a dimensionless constant that represents the frictional force between two particular surfaces. In this exercise, \( \mu \) affects how much force is required to move an object across a horizontal plane. A higher \( \mu \) indicates greater resistance to movement, requiring more force to drag the object.

• Coefficient of static friction: prevents motion from starting.
• Coefficient of kinetic friction: resists motion while sliding.

Here, we focus on kinetic friction, the resistance faced when the object is in motion. It shows how important \( \mu \) is to both understanding and optimizing the force required to keep something sliding steadily.

Minimization Problem

Solving a minimization problem means finding the point where a function reaches its lowest value. In this case, we are looking for the angle \( \theta \) that minimizes the force \( F \).

• We express \( F = \frac{\mu W}{\mu \sin \theta + \cos \theta} \), relying on trigonometric components to find the minimal force.
• The focus is on the denominator, \( \mu \sin \theta + \cos \theta \), since minimizing the denominator will also minimize \( F \).
• By taking the derivative \( D'(\theta) = \mu \cos \theta - \sin \theta \) and setting it to zero, we find the critical point \( \tan \theta = \mu \).
• Checking the second derivative \( D''(\theta) \) ensures that this critical point is a minimum.

This step-by-step method informs us on how to minimize expressions involving friction by strategically adjusting \( \theta \).

Trigonometry in Physics

Trigonometry allows us to analyze angles and relationships in physical scenarios, especially in calculating forces like in this exercise. Here, angles and trigonometric functions are essential:

• Understanding \( \sin \theta \) and \( \cos \theta \) is critical because they describe the components of the angle \( \theta \).
• Using \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) helps determine the optimal \( \theta \) that minimizes \( F \).
• Tying \( \theta \) to \( \mu \) directly relates friction properties with angular position, thereby optimizing motion control.

Thinking through these trigonometric relationships helps you figure out why certain angles simplify problems about forces in physics. This builds on practical skills in how angles impact real-world physical systems.