Question

A system consists of two identical cubes each of mass m

Answer 1

system consists of two identical cubes, each of mass m, linked together by the compressed weightless spring of stiffness χ (Fig. 1.41). The cubes are also connected by a thread which is burned through at a certain moment. Find: (a) at what values of Δl, the initial compression of the spring, the lower cube will bounce up after the thread has been burned through; (b) to what height h the centre of gravity of this system will rise if the initial compression of the spring Δl = 7 mg/χ.  

Solution:

A system consists of two identical cubes, each of mass m, linked together by the

Answer 2

Answer: 3mg/k.

Given: Initial compression = 7 MG/k

Solution:

The initial compression of the string ∆l is such that, after burning the thread, the cube which is at the top rises to a position which creates a tension on the spring which is at least same as that of the mass of the lower cube. In general, the spring first go to natural state from its compressed position then to the elongated position. Let “l” be the maximum elongation produced,

Therefore,

$kl = mg \rightarrow (1)$

According to laws of conservation of energy,

$\frac {1}{2} k \Delta l^2 = mg (\Delta l + l) +\frac {1}{2} kl^2\rightarrow(2)$

At maximum elongation, the speed of the upper cube = 0,

$\Delta l^2 - 2mg\Delta l/k - 3m^2g^2/k^2 = 0$

$\Delta l$ = (2mg ± 4mg)/ 2k

$\Delta l$ = (2mg + 4mg)/ 2k = 3mg/k

$\Delta l$ = (2mg - 4mg)/ 2k = -2mg/2k =  -mg/k

The possible solution is, 3mg/k.