Question

A block of mass m placed on a smooth floor is connected to
a fixed support with the help of a spring of force constant k.
It is pulled by a rope as shown in the figure. Tension T of the
rope is increased gradually without changing its direction,
until the block losses contact the floor. The increase in rope
tension T is so gradual that acceleration in the block can be
neglected.

(a) What is the necessary tension in the rope so that the block looses contact with the floor?
(b) What is the extension in the spring, when the block looses contact with the floor?​

Answer 1

Tension in the string has 2 components

Tsinx and Tcosx

where x is the angle at which block leaves contact

for the block to leave contact from the surface

weight of the block=Tsinx

mg=Tsinx

mg/sinx=T

extension in the spring = Tcosx

kx=Tcosx

x=Tcosx/k

=(mg/sinx)cosx/k

=mgcotx/k

Answer 2

The necessary tension in the rope so that the block loses contact with the floor is mg/sinθ

The extension in the string when the block loses contact with the floor is (mgcotθ)/k  

• The tension T generated in the string has two components namely Tcosθ and Tsinθ
• When the block loses contact with the floor sine component of tension becomes equal to weight of the block as Tsinθ=mg
• Therfore T=mg/sinθ
• when the block loses contact with the floor the the cosine component of tension becomes equal to the spring force 'kx' where x is the displacement and k is the spring constant as Tcosθ=kx
• Putting the value of T which result as x=(mgcotθ)/k