Question

Derive an expression for the acceleration of a body sliding down a rough inclined plane

Answer 1

$F_{n}$ = normal force on the body by the surface of incline

m = mass of the body

θ = angle of incline

μ = coefficient of kinetic friction

a = acceleration of the body

$f_{k}$ = kinetic frictional force

perpendicular to incline surface , force equation can be given as

$F_{n}$ = mg Cosθ                                  eq-1

kinetic frictional force is given as

$f_{k}$ = μ $F_{n}$

using eq-1

$f_{k}$ = μ mg Cosθ                              eq-2

parallel to incline , force equation is given as

mg Sinθ - $f_{k}$ = ma

using eq-2

mg Sinθ - μ mg Cosθ = ma

a = g (Sinθ -  μ Cosθ )

Answer 2

Answer:

f we neglect friction between the body and the plane - the force required to move the body up an inclined plane can be calculated as

Fp = W h / l

   = W sin α

   = m ag sin α                            (1)

where

Fp = pulling force (N, lbf)

W = m ag  

   =  gravity force - or weight of body (N, lbf)

h = elevation (m, ft)

l = length (m, ft)

α = elevation angle (degrees)

m = mass of body (kg, slugs)

ag = acceleration of gravity  (9.81 m/s2, 32.174 ft/s2)

By adding friction - (1) can be modified to

Fp = W (sin α + μ cos α)  

   = m ag (sin α + μ cos α)                            (2)

where

μ = friction coefficient

Example - Pulling Force on Inclined Plane

A body with mass 1000 kg is located on a 10 degrees inclined plane. The pulling force without friction can be calculated as

Fp = (1000 kg) (9.81 m/s2) sin(10o)

  = 1703 N

  = 1.7 kN

HOPE IT MAY HELPFUL