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

An object with weight $\mathrm{W}$ 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 $$\mathrm{F}=\frac{\mu \mathrm{W}}{\mu \sin \theta+\cos \theta}$$ 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 $\mathrm{W}$ 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
$$\mathrm{F}=\frac{\mu \mathrm{W}}{\mu \sin \theta+\cos \theta}$$
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

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.

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.

Transcript

00:01 Here we have a picture of a man who's pulling a package across the ground.

00:06 It's attached with the rope.

00:08 The formula we have is for force.

00:10 Force is equal to mu times w over mu sine of theta plus cosine of theta.

00:17 Then this formula, w represents the weight of the package.

00:22 Mu is the friction constant.

00:25 And theta is our angle between the ground and the rope.

00:30 So this gentleman's friend tells him to keep his force, at a minimum, you should keep the tangent of theta equal to mu.

00:39 So let's just work that out and see if that's true.

00:43 All right, if we think about that, force, to minimize our force, we'd have to see where the derivative of force is equal to zero.

00:54 Because if a derivative is equal to zero, that's either the minimum or the maximum point on our graph.

01:00 So we need to first find the derivative of our force.

01:06 To do that because we have a division problem here, we're going to have to implement the quotient rule.

01:14 The quotient rule says we have to take the denominator, so mu sine of theta plus cosine of theta, times the derivative of the numerator.

01:28 The derivative of mu w is just going to be zero because those are both constants.

01:35 We're going to subtract the numerator times the derivative of the denominator.

01:44 Well, the derivative of mu -sign of theta is going to be mu -cosine of theta, and the derivative of cosine of theta is going to be negative sign of theta.

01:59 And that's all over the denominator squared.

02:04 So mu sine of theta plus cosine of theta, the quantity of that squared...

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