An object, previously at rest, slides $3 m$ $3 m$ down a ramp, with an incline of $\frac{π}{8}$ $pi/8$ , and then slides horizontally on the floor for another $4 m$ $4 m$ . If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Question

An object, previously at rest, slides $3 m$ $3 m$ down a ramp, with an incline of $\frac{π}{8}$ $pi/8$ , and then slides horizontally on the floor for another $4 m$ $4 m$ . If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

1 Answer

Impact of this question

1348 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-22T07:02:54

The coefficient of friction of the material is $= 0.17$ $=0.17$

Explanation:

At the top of the ramp, the object possesses potential enegy.

At the botton of the ramp, part of the potential energy is converted into kinetic energy and the work done by the frictional force.

The length of the ramp is $s = 3 m$ $s=3m$

The angle of the incline is $θ = \frac{1}{8} π$ $theta=1/8pi$

Therefore,

$P E = m g h = m g \cdot s sin θ$ $PE=mgh=mg*ssintheta$

$μ_{k} = \frac{F_{r}}{N} = \frac{F_{r}}{m g cos θ}$ $mu_k=F_r/N=F_r/(mgcostheta)$

$F_{r} = μ_{k} \cdot m g cos θ$ $F_r=mu_k*mgcostheta$

Let the speed at the bottom of the ramp be $(u) m s^{- 1}$ $(u)ms^-1$

so,

$m g \cdot s sin θ = \frac{1}{2} m u^{2} + s \cdot μ_{k} \cdot m g cos θ$ $mg*ssintheta=1/2m u^2+s*mu_k*mgcostheta$

$g s sin θ = \frac{1}{2} u^{2} + s μ_{k} g cos θ$ $gssintheta=1/2u^2+smu_kgcostheta$

$u^{2} = 2 g s (sin θ - μ_{k} cos θ)$ $u^2=2gs(sintheta-mu_kcostheta)$

On the horizontal part,

The initial velocity is $= u$ $=u$

The final velocity is $v = 0$ $v=0$

The distance is $d = 4 m$ $d=4m$

The deceleration is calculated with Newton's Second Law of Motion

$F=F'_r=mu_kmg=ma$

$a=-mu_kg$

We apply the equation of motion

$v^2=u^2+2ad$

$0=2gs(sintheta-mu_kcostheta)-2mu_kgd$

$2mu_kgd=2gssintheta-mu_k2gscostheta$

$2mu_kgd+mu_k2gscostheta=2gssintheta$

$mu_k(2gd+2gscostheta)=2gssintheta$

The acceleration due to gravity is $g=9.8ms^-2$

$mu_k=(gssintheta)/(gd+gscostheta)$

$=(9.8*3*sin(1/8pi))/(9.8*4+9.8*3*cos(1/8pi))$

$=(11.25)/(66.36)$

$=0.17$

Science

Math

Humanities

... and beyond

An object, previously at rest, slides $3 m$ $3 m$ down a ramp, with an incline of $\frac{π}{8}$ $pi/8$ , and then slides horizontally on the floor for another $4 m$ $4 m$ . If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

An object, previously at rest, slides 3 m3m3 m down a ramp, with an incline of pi/8 π8pi/8 , and then slides horizontally on the floor for another 4 m4m4 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

1 Answer

Explanation:

Related questions

Impact of this question

An object, previously at rest, slides $3 m$ $3 m$ down a ramp, with an incline of $\frac{π}{8}$ $pi/8$ , and then slides horizontally on the floor for another $4 m$ $4 m$ . If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?