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

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

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Answer 1 · 2016-04-09T13:52:27

Here length of the ramp (l)= 5m

Angle of inclination of the ramp $θ = \frac{π}{3}$ $theta = pi/3$
Height of the object from horizontal floor, $h = l sin θ$ $h=lsintheta$
If mass of the body is $m k g$ $m kg$
The initial gravitational potential energy of the body $P E = m g h = m g l sin θ = 5 m g sin θ J$ $PE=mgh=mglsintheta=5mgsintheta J$

The normal reaction acting on the body when it is sliding down the ramp is $N_{r} = m g cos θ$ $N_r=mgcostheta$
and the corresponding frictional force $F_{r} = μ N_{r} = μ m g cos θ$ $F_r=muN_r=mumgcostheta$
where $μ =$ $mu=$ coefficient of kinetic friction
work done against frictional force when sliding down the ramp $W_{1} = F_{r} \times l = μ m g cos θ \times 5 J$ $W_1=F_rxxl=mumgcosthetaxx5J$

when the body slides on horizontal force,then normal reaction $N_{f} = m g$ $N_f=mg$ and corresponding frictional force $F_{f} = μ N_{f} = μ m g$ $F_f=muN_f=mumg$
work done against frictional force when sliding 6m along floor
$W_{2} = F_{f} \times 3 = 3 μ m g J$ $W_2=F_fxx3=3mumgJ$

Now applying conservation of mechanical energy we can write

The initial KE being zero
Initial PE = total work done against frictional force $= W_{1} + W_{2}$ $=W_1+W_2$
$:. 5mgsintheta=W_1+W_2$
$=>5mgsintheta=5mumgcostheta+3mumg$
$=>mu=(5sintheta)/(3+5costheta)=(5sin(pi/3))/(3+5cos(pi/3))$
$=(5xxsqrt3/2)/(3+5xx1/2)=(2.5xxsqrt3)/5.5=5/11xxsqrt3=~~0.79$

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

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