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Sign convention-Down the slope(-) and up the slope(+)
-For upward motion,
A = Inclination = 30o.
So, weight component acting down the slope = F1 = -mgsinA
Normal reaction = mgcosA
Coefficient of friction = u
So, frictional force(Acting down the slope) = F2 = -umgcosA
So net force = F1 + F2 = -mg(sinA + ucosA)
Acceleration = -g(sinA + ucosA) = a1
As accn is uniform, eqn of motion are applicable.
Let s = displacement up the slope
Launch velocity = u; ta = time of ascent and td=time of descent
So, ta = 1/2td [Acc. to qts]----------(1)
So, s = uta+ 1/2a1ta2.
or s = uta + 1/2[-g(sinA + ucosA)]ta2.----------(2)
Now, final velocity = v =0
Also, v = u + a1ta
or 0 = u + [-g(sinA + ucosA)]ta.
or u = g(sinA + ucosA)ta.
Put u = g(sinA + ucosA)ta in (2) we get,
s = g(sinA + ucosA)ta2 - 1/2[g(sinA + ucosA)]ta2.
or s = 1/2[g(sinA + ucosA)]ta2
or s = 1/2[g(sinA + ucosA)](1/2td)2 [From(1)]
or s =1/8[g(sinA + ucosA)]td2.--------------(3)
For downward motion,
Weight component(acting down the slope) = -mgsinA = F1
frictional force(acting up the slope) = umgcosA = F2.
So, net force = F1 + F2 = mg(-sinA + ucosA)
Accn = a2 = g(ucosA - sinA)
Displacement(Down the slope) = -s
time of descent = td.
So, -s = utd + 1/2a2td2
As the downard motion starts with u = 0, the above expression becomes,
-s = 1/2a2td2= 1/2[g(ucosA - sinA)]td2------------(4)
Adding (3) and (4) we get,
1/8[g(sinA + ucosA)]td2 + 1/2[g(ucosA - sinA)]td2 = 0
or1/8[g(sinA + ucosA)]td2 = -1/2[g(ucosA - sinA)]td2.
or1/8[g(sinA + ucosA)]td2= 1/2[g(sinA - ucosA)]td2.
or 4sinA - 4ucosA = sinA + ucosA
or u = 3/5tanA = 3/5tan(30o) [As A = 30o]
or u = (3)1/2/5
Hence, the coefficient of friction = (3)1/2/5
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