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

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

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

Key Concepts

Optimization in Calculus

This concept involves finding the minimum (or maximum) value of a function by taking its derivative, setting the derivative equal to zero, and solving for the variable. In physical problems, optimization is crucial to determine conditions that minimize energy, force, or cost, thereby simplifying complex systems to their most efficient state.

Trigonometric Functions and Their Derivatives

Trigonometric functions such as sine, cosine, and tangent play a key role in modeling relationships involving angles. In optimization problems, their derivatives help determine how changes in the angle affect the system, leading to conditions such as tan ? equaling a given constant. These relationships are essential for simplifying and solving problems involving angular dependencies.

Friction and Normal Force

In mechanics, friction is modeled as a force proportional to the normal force, where the coefficient of friction quantifies the proportionality. When forces are applied at an angle, the normal force is modified due to the vertical component of the applied force, affecting the frictional force. Understanding this interplay is fundamental when calculating the required force to overcome friction in practical applications.

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