Question

the ratio of acceleration of blocks A placed on smooth incline with block B on rough incline is 2 : 1.The coefficient of kinetic friction between blockB and incline is

Answer 1

The ratio of acceleration of blocks A placed on smooth incline with block B on rough incline is 2 : 1
We have to find the coefficient of kinetic friction between block B and incline.
For smooth surface, net force is
F = mg sin Θ
Since, F = ma 
So,
ma = mg sin Θ
a = g sin Θ
Here, Θ = 45°
a = g sin 45°
a = g / √ 2  ........ (1)
For rough surface,
F = mg sin Θ - F.r
F = mg sin Θ - μ mg cos Θ
m a' = mg (sin Θ - μ cos Θ)
a' = g (sin Θ - μ cos Θ)
a' = g  (sin 45° - μ cos 45°)
a' = g (1 / √ 2 - μ / √ 2)
Given that,
The ratio of acceleration is 2:1
So,
a / a' = 2 / 1
(g / √ 2) / (g (1 / √ 2 - μ / √ 2)) = 2
1 = 2 - 2 μ
2 μ = 2 - 1
2 μ = 1
μ = 1 / 2 = 0.5
So, the co-efficient of kinetic friction = μ = 0.5
This is the required answer. 

Answer 2

A smooth inclined plane should have θ=45 degrees
Therefore 1/2*tan 45 degrees= 1/2 (because tan 45 degrees=1)