How to find the coefficient of kinetic friction?

[audrey1203]

Homework Statement

A child slides down a slide with a 28° incline, and at the bottom her speed is precisely half what it would have been if the slide had been friction-less. Calculate the coefficient of kinetic friction between the slide and the child.
Knowns:
28°
speed=V/2

Homework Equations

$\Sigma F=ma$
${ F }_{ fr }={ \mu }_{ k }{ F }_{ N }$
${ v }^{ 2 }={ v }_{ 0 }^{ 2 }+2a(x-{ x }_{ 0 })$

The Attempt at a Solution

$\Sigma { F }_{ y }=ma=0$
${ F }_{ N }-mgcos\sigma =0$
${ F }_{ N }=mgcos\sigma$

${ \Sigma F }_{ x }=ma$
$-{ F }_{ fr }+mgsin\sigma =ma$
${ F }_{ fr }=\frac { ma-mgsin\sigma }{ -1 }$
${ F }_{ fr }=-ma+mgsin\sigma$

${ F }_{ fr }={ \mu }_{ k }{ F }_{ N }$
$-ma+mgsin\sigma ={ \mu }_{ k }mgcos\sigma$
$\frac {-ma+mgsin\sigma }{ mgcos\sigma } ={ \mu }_{ k }$
$\frac { -a+gsin\sigma }{ gcos\sigma } ={ \mu }_{ k }$

 

[TSny]

WELCOME TO PF!

$-ma+mgsin\sigma ={ \mu }_{ k }mgcos\sigma$
$\frac {-ma+mgsin\sigma }{ mgcos\sigma } ={ \mu }_{ k }$
$\frac { -a+gsin\sigma }{ gcos\sigma } ={ \mu }_{ k }$

OK. But you can see that solving for $\mu_k$ here does not seem to help much since you don't know the value of the acceleration $a$.
Instead of using the first equation above to solve for $\mu_k$, think of the equation as giving you the acceleration when friction is present.
What would this equation look like if there was no friction?

How must the acceleration with friction compare to the acceleration without friction in order for the final speed to be cut in half with friction?

 

[gneill]

Hi audrey1203, Welcome to Physics Forums!

One way to tackle the problem is to look at it in terms of kinematics: forces and accelerations, which is what you've shown. Another way is to consider an energy approach.

Since KE is proportional to the square of the velocity, and if the velocity with friction present is one half that of when there's no friction, then the KE with friction must be 1/4 that of the frictionless case. That is, $KE_o = 4 KE_f$, where $KE_o$ is the kinetic energy of the child at the bottom of the slide when no friction is present , and $KE_f$ is the kinetic energy when friction is present. You should confirm that for yourself by writing out the expressions for KE when the velocity is v and v/2 and comparing the results.

Hint: If you choose an arbitrary value for the length of the slide (maybe call it L) then you can determine the change gravitational PE during the trip and any energy loss due to friction since you know how to find the friction force. All the unknowns (like L and the mass of the child) should disappear in the algebra along the route