A mass is attached to two springs as shown in the figure. The spring constants of the two springs are and . For the frictionless surface, the time period of oscillation of mass is [2023]

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Q.
A mass $m$ is attached to two springs as shown in the figure. The spring constants of the two springs are $K_{1}$ and $K_{2}$ . For the frictionless surface, the time period of oscillation of mass $m$ is [2023]

1 $2 π \sqrt{\frac{m}{K_{1} + K_{2}}}$

2 $\frac{1}{2 π} \sqrt{\frac{K_{1} - K_{2}}{m}}$

3 $\frac{1}{2 π} \sqrt{\frac{K_{1} + K_{2}}{m}}$

4 $2 π \sqrt{\frac{m}{K_{1} - K_{2}}}$

Ans.
(1)

On displacing mass $m$ to the right by $x$ ,

$F = - (k_{1} x + k_{2} x) = - (k_{1} + k_{2}) x$

$a = \frac{F}{m} = - (\frac{k_{1} + k_{2}}{m}) x = - ω^{2} x$

$∴$ $ω = \sqrt{\frac{k_{1} + k_{2}}{m}} \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

Solution

Q.
A mass $m$ is attached to two springs as shown in the figure. The spring constants of the two springs are $K_{1}$ and $K_{2}$ . For the frictionless surface, the time period of oscillation of mass $m$ is [2023]

1 $2 π \sqrt{\frac{m}{K_{1} + K_{2}}}$

2 $\frac{1}{2 π} \sqrt{\frac{K_{1} - K_{2}}{m}}$

3 $\frac{1}{2 π} \sqrt{\frac{K_{1} + K_{2}}{m}}$

4 $2 π \sqrt{\frac{m}{K_{1} - K_{2}}}$

Ans.
(1)

On displacing mass $m$ to the right by $x$ ,

$F = - (k_{1} x + k_{2} x) = - (k_{1} + k_{2}) x$

$a = \frac{F}{m} = - (\frac{k_{1} + k_{2}}{m}) x = - ω^{2} x$

$∴$ $ω = \sqrt{\frac{k_{1} + k_{2}}{m}} \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$

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Solution

Q. A mass m is attached to two springs as shown in the figure. The spring constants of the two springs are K1 and K2. For the frictionless surface, the time period of oscillation of mass m is [2023]

1 2πmK1+K2

2 12πK1-K2m

3 12πK1+K2m

4 2πmK1-K2

Ans. (1) On displacing mass m to the right by x, F=-(k1x+k2x)=-(k1+k2)x a=Fm=-(k1+k2m)x=-ω2x ∴ ω=k1+k2m⇒T=2πω=2πmk1+k2

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Q.
A mass $m$ is attached to two springs as shown in the figure. The spring constants of the two springs are $K_{1}$ and $K_{2}$ . For the frictionless surface, the time period of oscillation of mass $m$ is [2023]

1 $2 π \sqrt{\frac{m}{K_{1} + K_{2}}}$

2 $\frac{1}{2 π} \sqrt{\frac{K_{1} - K_{2}}{m}}$

3 $\frac{1}{2 π} \sqrt{\frac{K_{1} + K_{2}}{m}}$

4 $2 π \sqrt{\frac{m}{K_{1} - K_{2}}}$

Ans.
(1)

On displacing mass $m$ to the right by $x$ ,

$F = - (k_{1} x + k_{2} x) = - (k_{1} + k_{2}) x$

$a = \frac{F}{m} = - (\frac{k_{1} + k_{2}}{m}) x = - ω^{2} x$

$∴$ $ω = \sqrt{\frac{k_{1} + k_{2}}{m}} \Rightarrow T = \frac{2 π}{ω} = 2 π \sqrt{\frac{m}{k_{1} + k_{2}}}$