An object with weight W is dragged along a horizontal plane...

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(s) of friction and where $0 \leqslant \theta \leqslant \pi / 2$. Show that $F$ is minimized when $\tan \theta=\mu$.

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(s) of friction and where $0 \leqslant \theta \leqslant \pi / 2$. Show that $F$ is minimized when $\tan \theta=\mu$.

Key Concepts

Coefficient of Friction

The coefficient of friction is a dimensionless constant that represents the ratio of the force of friction between two bodies and the normal force pressing them together. In the context of mechanics, it quantifies how easily one object will slide over another and appears in equations that model frictional forces.

Decomposition of Forces

Decomposing forces involves resolving a single force vector into its horizontal and vertical components. This technique allows us to analyze the effects of each component separately, particularly useful in scenarios where forces act at an angle, such as when a force is applied via a rope, influencing both the motion and the normal force on the object.

Normal Force and Its Modification

In problems involving friction on a horizontal plane, the normal force is typically equal to the object's weight. However, when an external force is applied at an angle, the vertical component of that force alters the effective normal force. Understanding this modification is crucial as it directly affects the frictional force experienced by the object.

Optimization Using Calculus

Optimization involves finding the conditions that minimize or maximize a given function. In mechanics, this often translates to finding the angle or configuration that minimizes the required applied force for a desired motion. This is typically achieved by taking the derivative of the function with respect to the variable of interest and setting it equal to zero, thereby finding the critical point.

Trigonometric Relationships in Force Analysis

Trigonometric functions, such as sine, cosine, and tangent, play a vital role in relating the components of forces acting at an angle. In this context, expressing the balance of forces and setting the derivative equal to zero leads to an optimal condition characterized by the relationship tan ? = ?, linking the angle of the applied force directly to the coefficient of friction.

Please give Ace some feedback