Q3.4-20E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

An object at rest on an inclined plane will not slide until the component of the gravitational force down the incline is sufficient to overcome the force due to static friction. Static friction is governed by an experimental law somewhat like that of kinetic friction (Problem 18); it has a magnitude of at most N, where m is the coefficient of static friction and N is, again, the magnitude of the normal force exerted by the surface on the object. If the plane is inclined at an angle a, determine the critical value ₀ for which the object will slide if $\mathit{a}\mathbf{>}\mathit{a}$_o   but will not move for $\mathit{a}\mathbf{<}\mathit{a}$_o.

Step-by-Step Solution

Answer

Therefore, the object starts sliding down when ${\mathbf{tan\alpha }}_{\mathbf{o}}\mathbf{>}\mathbf{\mu }$.

Step 1: Draw a diagram

Step 2: Find the critical value

Here C is the center of mass of an object. As the object starting sliding down 

 

$\left|{\mathbf{mgsin\alpha }}_{\mathbf{o}}\right|\mathbf{>}\left|\mathbf{\mu N}\right|$

 

However,  $\left|\mathbf{\mu N}\right|\mathbf{=}\left|{\mathbf{\mu mgcos\alpha }}_{\mathbf{o}}\right|$

 

Hence the object starts sliding down when $\left|{\mathbf{mgsin\alpha }}_{\mathbf{o}}\right|\mathbf{>}\left|{\mathbf{\mu mgcos\alpha }}_{\mathbf{o}}\right|$

 $\left|{\mathbf{sin\alpha }}_{\mathbf{o}}\right|\mathbf{>}\left|{\mathbf{\mu cos\alpha }}_{\mathbf{o}}\right|$

 

Assume that the inclined plain is not vertical. Hence, $\cos {\alpha }_o\ne 0$ . Then

 

$\left|\frac{{\mathbf{sin\alpha }}_{\mathbf{o}}}{{\mathbf{cos\alpha }}_{\mathbf{o}}}\right|\mathbf{=}\left|{\mathbf{tan\alpha }}_{\mathbf{o}}\right|$

 

Both $\mathbf{\mu }$and ${\mathbf{tan\alpha }}_{\mathbf{o}}$are positive, 

so $\left|{\mathbf{tan\alpha }}_{\mathbf{o}}\right|\mathbf{=}{\mathbf{tan\alpha }}_{\mathbf{o}}$and $\left|\mathbf{\mu }\right|\mathbf{=}\mathbf{\mu }$

 

Therefore, the object starts sliding down when ${\mathbf{tan\alpha }}_{\mathbf{o}}\mathbf{>}\mathbf{\mu }$.