# An object with a mass of  3  kg is lying on a surface and is compressing a horizontal spring by 10  cm. If the spring's constant is  9 kgs^-2, what is the minimum value of the surface's coefficient of static friction?

Apr 7, 2016

The frictional coefficient $\mu = 0.03$.

#### Explanation:

(convert the distance to SI units: $10$ $c m$ = $0.1$ $m$)

The force exerted by the spring is given by:

$F = k x = 9 \times 0.1 = 0.9$ $N$

Since the object is stationary, this force applied by the spring must be being balanced by the frictional force.

The frictional force is given by:

${F}_{\text{frict"=muF_"normal}}$

And the normal force is the gravitational force:

${F}_{\text{normal}} = m g = 3 \times 9.8 = 29.4$ $N$

Rearranging:

$\mu = {F}_{\text{frict"/F_"normal}} = \frac{0.9}{29.4} = 0.03$

(frictional coefficients do not have units)

(side note: I keep saying that $k g {s}^{-} 2$ is an odd unit for spring constant. It is correct, but so is $N {m}^{-} 1$, and the latter makes is easier to understand what is going on in most contexts.)