Question

Two blocks of mass m₁ and m₂ joined to a non deformed spring of length l₀ and stiffness K as shown in fig. If a force F is applied on block of mass m₂. Find the maximum separation between the blocks.

• A. l₀ + $\frac { \mathrm { m } _ { 1 } \mathrm {~F} } { \mathrm { k } \left( \mathrm { m } _ { 1 } + \mathrm { m } _ { 2 } \right) }$
• B. l₀ + $\frac { m _ { 2 } \mathrm {~F} } { \mathrm { k } \left( \mathrm { m } _ { 1 } + \mathrm { m } _ { 2 } \right) }$
• C. l₀ + $\frac { 2 \mathrm {~m} _ { 2 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } }$
• D. l₀ + $\frac { 2 \mathrm {~m} _ { 1 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } }$

Answer 1

Answer: (D)

l₀ + $\frac { 2 \mathrm {~m} _ { 1 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } }$

Answer 2

x₁ and x₂ be the maximum displacement in m₁ and m₂ respectively.

a_COM = $\frac { \mathrm { F } } { \mathrm { m } _ { 1 } + \mathrm { m } _ { 2 } }$

$\frac { 1 } { 2 } \mathrm { k } \left( \mathrm { x } _ { 1 } + \mathrm { x } _ { 2 } \right) ^ { 2 } = \frac { \mathrm { m } _ { 1 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } } \mathrm { x } _ { 1 } + \left( \mathrm { F } - \frac { \mathrm { m } _ { 2 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } } \right) \mathrm { x } ^ { 2 }$

= $\frac { \mathrm { m } _ { 1 } \mathrm {~F} } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } }$ (x₁ + x₂)

or

x₁ + x₂ = $\frac { 2 m _ { 1 } F } { k \left( m _ { 1 } + m _ { 2 } \right) }$

Thus maximum separation = l₀ + x₁ + x₂

= l₀ + $\frac { 2 m _ { 1 } F } { k \left( m _ { 1 } + m _ { 2 } \right) }$