Question

An object, previously at rest, slides `5 m` down a ramp, with an incline of `pi/3`, and then slides horizontally on the floor for another `2 m`. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Answer 1

(I really want someone to double check my work since this is one of my first physics answers)

`mu_k = 0.9622`

Explanation:

Let's use the top of the ramp for initial condition and bottom of the ramp for final condition.

`PE_i + KE_i = PE_f + KE_f + W_"fric"`

We know that:

`Deltay = 5m(sin(pi/3)) = (5sqrt3)/2`
`KE_i = 0`
`PE_i = mgDeltay = (5sqrt3mg)/2`
`PE_f = 0`
`W_"fric" = 5mu_kF_N = 5mu_kmgcos(pi/3)= 5/2mu_kmgcolor(white)"-"`

So our equation becomes:

`(5sqrt3mg)/2 = KE_f + 5/2mu_kmg`

We can also have another point (`f2`) at the end of the 2 meters when the block is at rest.

`PE_f + KE_f = PE_"f2" + KE_"f2" + W_"fric2"`

`PE_f, PE_"f2", and KE_"f2"` are all 0, and `W_"fric2" = 2mu_kmg`

`KE_f = 2mu_kmg`

Substituting this into the first equation gives:

`(5sqrt3mg)/2 = 2mu_kmg + 5/2mu_kmg`

`(5sqrt3)/2 = 9/2mu_k`

`mu_k = (5sqrt3)/9 = 0.9622`

Final Answer