A sonometer wire of resonating length 90 cm has a fundamental frequency of 400 Hz when kept under some tension. The resonating length of the wire with fundamental frequency of 600 Hz under same tension _______ cm.
Correct Answer: 60
Approach Solution - 1
Given:
- Initial resonating length, \( L = 90 \, \text{cm} \)
- Initial fundamental frequency, \( f_0 = 400 \, \text{Hz} \)
- New fundamental frequency, \( f' = 600 \, \text{Hz} \)
Step 1: Relation Between Frequency and Length
The fundamental frequency of a vibrating string is given by:
\[ f_0 = \frac{v}{2L}, \]
where:
- \( v \) is the wave speed,
- \( L \) is the length of the wire.
For the same tension, the wave speed \( v \) remains constant.
Step 2: Expressing New Length in Terms of Frequency
Let the new resonating length be \( L' \) for the frequency \( f' \). The new fundamental frequency is given by:
\[ f' = \frac{v}{2L'}. \]
Dividing the two equations:
\[ \frac{f'}{f_0} = \frac{L}{L'}. \]
Rearranging to find \( L' \):
\[ L' = L \times \frac{f_0}{f'}. \]
Step 3: Substituting the Given Values
Substituting the values:
\[ L' = 90 \times \frac{400}{600}. \]
Simplifying:
\[ L' = 90 \times \frac{2}{3} = 60 \, \text{cm}. \]
Therefore, the new resonating length of the wire is \( 60 \, \text{cm} \).
Approach Solution -2
Initial resonating length, l₁ = 90 cm
Initial frequency, f₁ = 400 Hz
New frequency, f₂ = 600 Hz
Tension and mass per unit length remain constant.
Step 2: Relation between frequency and resonating length.
For a sonometer wire under constant tension:
\[ f \propto \frac{1}{l} \] Therefore, \[ \frac{f₁}{f₂} = \frac{l₂}{l₁} \]
Step 3: Substitute the given values.
\[ \frac{400}{600} = \frac{l₂}{90} \] \[ l₂ = 90 \times \frac{400}{600} = 90 \times \frac{2}{3} = 60 \, \text{cm} \]
Step 4: Final Answer.
The new resonating length of the wire is:
\[ \boxed{60 \, \text{cm}} \]
Final Answer: 60 cm
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Waves Questions
- A closed organ pipe 150 cm long gives 7 beats per second with an open organ pipe of length 350 cm, both vibrating in fundamental mode. The velocity of sound is ________ m/s.
- The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60 cm, the length of the closed pipe will be :
- Two waves of intensity ratio \( 1 : 9 \) cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is \( I_1 \) (b) Waves are coherent is \( I_2 \) and differ in phase by \( 60^\circ \) If \( \frac{I_1}{I_2} = \frac{10}{x} \) then \( x = \) _____.
- The equation of a stationary wave is:
\[y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right)\]Which of the following is NOT correct: - A plane progressive wave is given by \( y = 2 \cos 2\pi(330t - x) \, \text{m} \). The frequency of the wave is:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.