# An object with a mass of 12 kg is on a surface with a kinetic friction coefficient of  1 . How much force is necessary to accelerate the object horizontally at  14 m/s^2?

Apr 20, 2016

$290 N \left[\text{forward}\right]$

#### Explanation:

For this problem it is important to note that there are two forces acting on the object in the $x$ direction, the applied and friction forces.

Start by listing out the given and required values.

m=12kgcolor(white)(i)color(teal),color(white)(i)mu_k=1color(white)(i)color(teal),color(white)(i)a=14m/s^2color(white)(i)color(teal),color(white)(i)F_"app"=?

A free-body diagram can represent the situation as:

In this case, set the positive directions to be forward and down.

Using ${F}_{\text{net} , y} = m {a}_{y}$, add the forces in the $y$ direction.

${F}_{\text{net} , y} = m {a}_{y}$

${F}_{N} + F g = m {a}_{y}$

Since the object does not accelerate vertically, the acceleration in the $y$ direction is $0 \frac{m}{s} ^ 2$.

${F}_{N} + F g = m \left(0 \frac{m}{s} ^ 2\right)$

${F}_{N} + F g = 0 N$

${F}_{N} = - F g$

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{F}_{N} = - m g} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

After adding the forces in the $y$ direction, do the same for the forces in the $x$ direction. Rearrange for ${F}_{\text{app}}$ to find the force needed to accelerate the object horizontally.

${F}_{\text{net} , x} = m {a}_{x}$

${F}_{\text{app}} + {F}_{f , k} = m {a}_{x}$

${F}_{\text{app}} = m {a}_{x} - {F}_{f , k}$

${F}_{\text{app}} = m {a}_{x} - {\mu}_{k} {F}_{N}$

${F}_{\text{app}} = m {a}_{x} - {\mu}_{k} \left(\textcolor{b l u e}{- m g}\right)$

${F}_{\text{app}} = m {a}_{x} + {\mu}_{k} m g$

${F}_{\text{app}} = \left(12 k g\right) \left(14 \frac{m}{s} ^ 2\right) + \left(1\right) \left(12 k g\right) \left(9.81 \frac{m}{s} ^ 2\right)$
${F}_{\text{app}} = 285.72 N$
${F}_{\text{app}} \approx \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} 290 N \textcolor{w h i t e}{\frac{a}{a}} |}}}$