An object with a mass of $5 k g$ $5 kg$ is on a plane with an incline of $- \frac{5 π}{12}$ $-(5 pi)/12$ . If it takes $12 N$ $12 N$ to start pushing the object down the plane and $2 N$ $2 N$ to keep pushing it, what are the coefficients of static and kinetic friction?

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An object with a mass of $5 k g$ $5 kg$ is on a plane with an incline of $- \frac{5 π}{12}$ $-(5 pi)/12$ . If it takes $12 N$ $12 N$ to start pushing the object down the plane and $2 N$ $2 N$ to keep pushing it, what are the coefficients of static and kinetic friction?

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Answer 1 · 2017-07-22T04:30:18

If angle of inclination is $\frac{π}{12}$ $pi/12$ :

$μ_{s} = 0.521$ $mu_s = 0.521$

$μ_{k} = 0.310$ $mu_k = 0.310$

(see explanation regarding angle)

Explanation:

We're asked to find the coefficient of static friction $μ_{s}$ $mu_s$ and the coefficient of kinetic friction $μ_{k}$ $mu_k$ , with some given information.

![upload.wikimedia.org]( useruploads.socratic.org )

We'll call the positive $x$ $x$ -direction up the incline (the direction of $f$ $f$ in the image), and the positive $y$ $y$ direction perpendicular to the incline plane (in the direction of $N$ $N$ ).

There is no net vertical force, so we'll look at the horizontal forces (we WILL use the normal force magnitude $n$ $n$ , which is denoted $N$ $N$ in the above image).

We're given that the object's mass is $5$ $5$ $kg$ $"kg"$ , and the incline is $- \frac{5 π}{12}$ $-(5pi)/12$ .

Since the angle is $- \frac{5 π}{12}$ $-(5pi)/12$ ( $- 75^{o}$ $-75^"o"$ ), this would be the angle going down the incline (the topmost angle in the image above). Therefore, the actual angle of inclination is

$\frac{π}{2} - \frac{5 π}{12} = \frac{π}{12}$ $pi/2 - (5pi)/12 = pi/12$

Which is much more realistic considering the object remains stationary (due to static friction) unless a force acts on it.

The formula for the coefficient of static friction $μ_{s}$ $mu_s$ is

$f_{s} \geq μ_{s} n$ $f_s >= mu_sn$

Since the object in this problem "breaks loose" and the static friction eventually gives way, this equation is simply

$f_{s} = μ_{s} n$ $f_s = mu_sn$

Since the two vertical quantities $n$ $n$ and $m g cos θ$ $mgcostheta$ are equal,

$n = m g cos θ = (5 l kg) (9.81 l {m/s}^{2}) cos (\frac{π}{12}) = 47.4$ $n = mgcostheta = (5color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)cos(pi/12) = 47.4$ $N$ $"N"$

Since $12$ $12$ $N$ $"N"$ is the "breaking point" force that causes it to move, it is this value plus $m g sin θ$ $mgsintheta$ that equals the upward static friction force $f_{s}$ $f_s$ :

$f_{s} = m g sin θ + 12$ $f_s = mgsintheta + 12$ $N$ $"N"$

$= (5 l kg) (9.81 l {m/s}^{2}) sin (\frac{π}{12}) + 12 l N = 24.7$ $= (5color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)sin(pi/12) + 12color(white)(l)"N" = 24.7$ $N$ $"N"$

The coefficient of static friction is thus

$mu_s = (f_s)/n = (24.7cancel("N"))/(47.4cancel("N")) = color(red)(0.521$

The coefficient of kinetic friction $mu_k$ is given by

$f_k = mu_kn$

It takes $2$ $"N"$ of applied downward force (on top of weight) to keep the object accelerating constantly downward, then we have

$f_k = mgsintheta + 2$ $"N"$

$= 12.7color(white)(l)"N" + 2$ $"N"$ $= 14.7$ $"N"$

The coefficient of kinetic friction is thus

$mu_k = (f_k)/n = (14.7cancel("N"))/(47.4cancel("N")) = color(blue)(0.310$

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An object with a mass of $5 k g$ $5 kg$ is on a plane with an incline of $- \frac{5 π}{12}$ $-(5 pi)/12$ . If it takes $12 N$ $12 N$ to start pushing the object down the plane and $2 N$ $2 N$ to keep pushing it, what are the coefficients of static and kinetic friction?

1 Answer

Explanation:

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An object with a mass of 5 kg5kg5 kg is on a plane with an incline of -(5 pi)/12 −5π12 -(5 pi)/12 . If it takes 12 N12N12 N to start pushing the object down the plane and 2 N2N2 N to keep pushing it, what are the coefficients of static and kinetic friction?

1 Answer

Explanation:

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An object with a mass of $5 k g$ $5 kg$ is on a plane with an incline of $- \frac{5 π}{12}$ $-(5 pi)/12$ . If it takes $12 N$ $12 N$ to start pushing the object down the plane and $2 N$ $2 N$ to keep pushing it, what are the coefficients of static and kinetic friction?