An object with a mass of $3 k g$ $3 kg$ is on a ramp at an incline of $\frac{π}{12}$ $pi/12$ . If the object is being pushed up the ramp with a force of $2 N$ $2 N$ , what is the minimum coefficient of static friction needed for the object to remain put?

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An object with a mass of $3 k g$ $3 kg$ is on a ramp at an incline of $\frac{π}{12}$ $pi/12$ . If the object is being pushed up the ramp with a force of $2 N$ $2 N$ , what is the minimum coefficient of static friction needed for the object to remain put?

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Answer 1 · 2017-08-02T22:48:35

$μ_{s} \geq 0.198$ $mu_s >= 0.198$

Explanation:

We're asked to find the minimum coefficient of static friction $μ_{s}$ $mu_s$ .

For this situation, the net horizontal force $\sum F_{x}$ $sumF_x$ is given by

$\sum F_{x} = m g sin θ - f_{s} - F_{applied} = 0$ $sumF_x = mgsintheta - f_s - F_"applied" = 0$

where

$F_{applied}$ $F_"applied"$ is $2$ $2$ $N$ $"N"$ (directed up the incline)
$f_{s}$ $f_s$ is the maximum static friction force (directed up the incline. Even though it is being pushed upward, the net force points downward in the absence of friction), which is given by

$f_s = mu_sn = mu_soverbrace(mgcostheta)^n$

So

$sumF_x = mgsintheta - mu_smgcostheta - 2$ $"N"$ $= 0$

Now we solve the expression for the coefficient of static friction $mu_s$ :

$mu_smgcostheta = mgsintheta - 2$ $"N"$

$ul(mu_s = (mgsintheta - 2color(white)(l)"N")/(mgcostheta)$

We know:

$m = 3$ $"kg"$
$g = 9.81$ $"m/s"^2$
$theta = pi/12$

Plugging these in gives

$mu_s = ((3color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)sin(pi/12) - 2color(white)(l)"N")/((3color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)cos(pi/12)) = color(blue)(ulbar|stackrel(" ")(" "0.198" ")|)$

The minimum coefficient of static friction is thus $color(blue)(0.198$ .

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An object with a mass of $3 k g$ $3 kg$ is on a ramp at an incline of $\frac{π}{12}$ $pi/12$ . If the object is being pushed up the ramp with a force of $2 N$ $2 N$ , what is the minimum coefficient of static friction needed for the object to remain put?

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An object with a mass of 3 kg3kg3 kg is on a ramp at an incline of pi/12 π12pi/12 . If the object is being pushed up the ramp with a force of 2 N2N 2 N, what is the minimum coefficient of static friction needed for the object to remain put?

1 Answer

Explanation:

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An object with a mass of $3 k g$ $3 kg$ is on a ramp at an incline of $\frac{π}{12}$ $pi/12$ . If the object is being pushed up the ramp with a force of $2 N$ $2 N$ , what is the minimum coefficient of static friction needed for the object to remain put?