# An object with a mass of  3 kg is lying on a surface and is compressing a horizontal spring by 20 cm. If the spring's constant is  9 (kg)/s^2, what is the minimum value of the surface's coefficient of static friction?

Sep 3, 2017

The minimum value of the coefficient of static friction is 0.06.

#### Explanation:

I assume that it is to be understood that you are asking for minimum value of the surface's coefficient of static friction that will prevent the spring from moving the mass.

The spring exerts a horizontal force of
${F}_{\text{spring" = k * Deltax = 9 "kg"/s^2 * 0.2 m = 1.8" kg}} \cdot \frac{m}{s} ^ 2 = 1.8 N$

Friction needs to be able to oppose that force with an equal and opposite force to maintain its grip. The force of friction is
${F}_{f} = {\mu}_{s} \cdot N$
where N is the component if the object's weight normal to the surface. I assume the surface is horizontal, so N is the object's full weight.
${F}_{f} = {\mu}_{s} \cdot m \cdot g$
$1.8 N = {\mu}_{s} \cdot 3 k g \cdot \frac{9.8 m}{s} ^ 2 = {\mu}_{s} \cdot \frac{29.4 k g \cdot m}{s} ^ 2 = {\mu}_{s} \cdot 29.4 N$
Solving for ${\mu}_{s}$,
${\mu}_{s} = \frac{1.8 \cancel{N}}{29.4 \cancel{N}} = 0.06$

I hope this helps,
Steve