Question

A system is shown in the figure. The time period for small oscillations of the two blocks will be -

• A. 2p
• B. 2p $\sqrt { \frac { 3 \mathrm {~m} } { 4 \mathrm { k } } }$
• C. 2p
• D. 2p $\sqrt { \frac { 3 \mathrm {~m} } { 2 \mathrm { k } } }$

Answer 1

Answer: (B)

2p $\sqrt { \frac { 3 \mathrm {~m} } { 4 \mathrm { k } } }$

Answer 2

Both the spring are in series

K_eq = $\frac { 2 \mathrm { k } } { 3 }$

Time period T = 2p $\sqrt { \frac { \mu } { \mathrm { K } _ { \mathrm { eq } } } }$

where µ = $\frac { \mathrm { m } _ { 1 } \cdot \mathrm { m } _ { 2 } } { \mathrm {~m} _ { 1 } + \mathrm { m } _ { 2 } }$ Here µ = $\frac { \mathrm { m } } { 2 }$

\ T = 2p = 2

Alternative method :

\ mx₁ = mx₂ Ž x₁ = x₂

force equation for first block;

$\frac { 2 \mathrm { k } } { 3 }$ (x₁+x₂) = m

Put x₁ = x₂ Ž $\frac { 4 \mathrm { k } } { 3 \mathrm {~m} }$ x₁ = 0

Ž w² = $\frac { 4 \mathrm { k } } { 3 \mathrm {~m} }$

\ T = 2p