# An object with a mass of  15 kg is lying on a surface and is compressing a horizontal spring by 10 cm. If the spring's constant is  8 (kg)/s^2, what is the minimum value of the surface's coefficient of static friction?

Feb 17, 2017

The minimum coefficient of static friction is $\mu = 5.4 \cdot {10}^{-} 3$.

#### Explanation:

We know that the frictional force due to static friction will be the normal force of the object on the surface times the coefficient of static friction, $\mu$. The normal force, in turn, is simply the block's mass times the force of gravity:

$F = - \mu N$
$F = - g m \mu$

And Hooke's law tells us that the force due to the spring is...

$F = - k x$

These two values will be equal when $\mu$ is at the minimum to keep the block stationary, so we can set the equations equal and plug in given values.

$- k x = - g m \mu$
$k x = g m \mu$
$8 \cdot .1 = 9.8 \cdot 15 \cdot \mu$
$.8 = 147 \mu$
$0.0054 = \mu$

And using scientific notation to simplify we get:

$\mu = 5.4 \cdot {10}^{-} 3$