At t = 0, a disk of radius 1 m starts rolling without slipping with constant angular acceleration α = 2/3 rad s⁻². A small stone is initially at the contact point. At t = √π s the stone detaches and flies tangentially. The maximum height measured from the plane is 1/2 + x/10 metres. Find x. Take g = 10 m s⁻².
Physics · Rotational Motion · combined translatory and rotatory motion ,Pure rolling motion
JEE Advanced 2022

Question

At t = 0, a disk of radius 1 m starts rolling without slipping with constant angular acceleration α = 2/3 rad s⁻². A small stone is initially at the contact point. At t = √π s the stone detaches and flies tangentially. The maximum height measured from the plane is 1/2 + x/10 metres. Find x. Take g = 10 m s⁻².

Physics · Rotational Motion · combined translatory and rotatory motion ,Pure rolling motion

JEE Advanced 2022

G1 Physics · Accepted Answer

For rolling from rest with constant angular acceleration α=2/3 rad s⁻² and R=1 m, at t=√π s the angular displacement is θ=½αt²=π/3 and angular speed is ω=αt=(2/3)√π. The stone leaves tangentially from the disc; resolve its tangential velocity into horizontal and vertical components, then use H=y₀+(vertical speed)²/(2g). This gives x=0.52.