A fixed wedge with both surface inclined at 45 to the horizontal as shown in the figure A particle P - Physics - Laws Of Motion

Question

A fixed wedge with both surface inclined at 45 to the horizontal as
shown in the figure A particle P of mass m is held on the smooth
plane by a light string which passes over a smooth pulley A and
attached to a particle Q of mass 3m which rest on the rough plane
The system is released from rest Give that the acceleration of each
particle is of magnitude g/5 root 2 then the tension in the string
is

Somnath Mazumder · Accepted Answer

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Let the coefficient of friction between the
block and the rough side = µ
Here,
a = g/(5√2)
By the FBD
f = 3µmgcosθ
3mgsin (450) – 3µmgcos (450)
– T = 3ma -----------1
T – mgsin (450) = ma ----------2
Adding eqn 1 and eqn 2:
2mgsin (450) – 3µmgcos (450) =
4ma = 4mg/(5√2)
=> 2 – 3µ = 4/5
=> µ = 0 4
Putting value of µ in eqn 1
3mgsin (450) – 1 2mgcos (450) – T
= 3mg/(5√2)
=> 3mg – 1 2mg – T = 0 6mg
=> T = 1 2mg = 6mg /5 -----3
Eqn 3 gives the amount of tension in the rope