Question

The system shown in the diagram comprises of two masses connected by a massless rigid rod of length LL. This rod, which can freely rotate about its hinge, is hinged at the blue block at one end. The other end of the rod is attached to a bob, as shown in the diagram. The blue block is constrained to move along a fixed horizontal surface. The spring is fixed at one end to a vertical wall and the other end is attached to the blue block. The goal of this problem is to analyse small vibrations about the equilibrium configuration of this system. Enter your answer as the sum of squares of the natural frequencies of small oscillations about the stable equilibrium point.

Note:

m₁ = m₂ = 2 ; K = 5 ; L₀ = L=1 ; g = 10m

L₀ is the natural length of the spring.

Treat all masses as point objects.

The bob's motion and that of the blue block are confined to the same plane.

Hint: Find the equations of motion of this system and evaluate the configuration at which the system is in stable equilibrium. Then, linearise the equations about the equilibrium point, and proceed.

Answer 1

$\huge\underline{\mathbb\red{❥︎A}\green{N}\mathbb\blue{S}\purple{W}\mathbb\orange{E}\pink{R}}\:$

The weight of lump is given as,

W .1 =mg

The volume of a lump is given as,

V= ρ ll

The weight of lump in brine solution is given as,

W 2=mg− 2ρgV

=mg(1− 2ρ !ρ )

The new readings of the spring balance is given as,W

=mg(1− 2ρ lρρl )

=200×(1− 2×11.41.1 )

=190.35gF

Thus, the new reading of the spring balance is 190.35gF.