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## Homework Statement

Assume a box is sliding down a ramp with an incline of [tex]\theta[/tex] radians and reaches terminal velocity before arriving at the bottom of the ramp. Assume that the drag force caused by the air is proportional to [tex]Sv^2[/tex], where [tex]S[/tex] is the cross sectional area perpendicular to the direction of motion and [tex]v[/tex] is the speed. Further assume that the sliding friction between the object and the ramp is proportional to the normal weight of the object. Determine the relationship between the terminal velocity and the mass of the object.

## The Attempt at a Solution

The forces acting on the box will be the drag force [tex]F_d[/tex] acting in the negative direction, the sliding friction [tex]F_s[/tex] acting in the negative direction, and the component of gravity that is parallel to the surface of the ramp [tex]F_g_x[/tex] acting in the positive direction. At terminal velocity, the box is not accelerating. Using Newton's Second Law for the net force acting on the box, we have

[tex] -F_s-F_d+F_g_x=0 [/tex]

**(1)**

Now, [tex]F_s \propto w [/tex] and [tex]w \propto m[/tex] so

[tex]F_s \propto m[/tex]

where [tex]w[/tex] is the normal weight of the box and [tex]m[/tex] is the mass of the box.

The same argument can be made for [tex]F_g_x[/tex]. So

[tex]F_g_x \propto m[/tex].

For [tex]F_d[/tex], note that [tex]S \propto L^2 [/tex] and [tex]V \propto L^3 [/tex],

where [tex]L[/tex] is length and [tex]V[/tex] is volume.

So [tex]L\propto S^{\frac{1}{2}} \propto V^{\frac{1}{3}}[/tex].

This implies that [tex]S \propto V^{\frac{2}{3}}[/tex].

Since volume [tex]V[/tex] is proportional to mass [tex]m[/tex], we have

[tex]S \propto m^{\frac{2}{3}}[/tex].

Assuming we are at terminal velocity, [tex]v=v_T[/tex].

Now going back to (1), we have

[tex] -F_s-F_d+F_g_x=0 [/tex]

[tex]-m-m^{\frac{2}{3}}v^2_{T}+m \propto 0 [/tex]

[tex]m^{\frac{2}{3}}v^2_{T} \propto 0[/tex]

**(2)**

The proportionality in (2) doesn't make sense. Where am I going wrong? Do I need to modify

[tex]F_g_x \propto m[/tex] to be [tex]F_g_x \propto \sin{(\theta)}m[/tex].?

Thanks

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