Question

When a mass m is connected individually to two springs S1 to S2, the oscillation frequencies are and . If the same mass is attached to the two springs as shown in figure, the oscillation frequency would be

• A. 1 + 2
• B. $\sqrt { v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } }$
• C. $\left( \frac { 1 } { v _ { 1 } } + \frac { 1 } { v _ { 2 } } \right) ^ { - 1 }$
• D. $\sqrt { v _ { 1 } ^ { 2 } - v _ { 2 } ^ { 2 } }$

Answer 1

Answer: (B)

$\sqrt { v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } }$

Answer 2

Let $\mathrm { k } _ { 1 }$ and $\mathrm { k } _ { 2 }$ be the spring constant of springs and respectively. Then

$v _ { 1 } = \frac { 1 } { 2 \pi } \sqrt { \frac { \mathrm { k } _ { 1 } } { \mathrm {~m} } }$

And

If k is effective spring constant of two springs $\mathrm { S } _ { 1 }$ and $\mathrm { k } = \mathrm { k } _ { 1 } + \mathrm { k } _ { 2 }$ ( springs are connected in parallel) if U is the effective frequency of oscillation when the mass m is attached to the sprins and $\mathrm { S } _ { 2 }$ as shown in figure. Then

$v = \frac { 1 } { 2 \pi } \sqrt { \frac { k } { m } } = \frac { 1 } { 2 \pi } \sqrt { \frac { k _ { 1 } + k _ { 2 } } { m } } = \frac { 1 } { 2 \pi } \sqrt { \frac { k _ { 1 } } { m } + \frac { k _ { 2 } } { m } }$

$v = \frac { 1 } { 2 \pi } \sqrt { 4 \pi ^ { 2 } v _ { 1 } ^ { 2 } + 4 \pi ^ { 2 } v _ { 2 } ^ { 2 } }$ (Using (1) and (ii))

$= \sqrt { v _ { 1 } ^ { 2 } + v _ { 2 } ^ { 2 } }$