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PHXI14:OSCILLATIONS

364153 A heavy small-sized sphere is suspended by a string of length $l$ . The sphere rotates uniformly in a horizontal circle with the string making an angle $θ$ with the vertical. Then, the time period of this conical pendulum is

1

t = 2 π \sqrt{\frac{l}{g}}

2

t = 2 π \sqrt{\frac{l \sin θ}{g}}

3

t = 2 π \sqrt{\frac{l \cos θ}{g}}

4

t = 2 π \sqrt{\frac{l}{g \cos θ}}

Explanation:

Radius of circular path in the horizontal plane
$Δ D A B, r = l \sin θ$
Resolving $T$ along the vertical and horizontal directions, we get
$\begin{matrix} (1) & T \cos θ = M g \end{matrix}$
supporting img
$T \sin θ = M r ω^{2} = M (l \sin θ) ω^{2}$
$\begin{matrix} (2) & \Rightarrow T = M l ω^{2} \end{matrix}$
On dividing Eq. (2) by Eq. (1), we get
$\frac{1}{\cos θ} = \frac{l ω^{2}}{g} \Rightarrow ω^{2} = \frac{g}{l \cos θ}$
$∴$ Time period, $t = \frac{2 π}{ω} = 2 π \sqrt{\frac{l \cos θ}{g}}$

PHXI14:OSCILLATIONS

364154 In a linear Simple Harmonic Motion ( $S H M$ )
(A) Restoring force is directly proportion to the displacement.
(B) The acceleration and displacement are opposite in direction.
(C) The velocity is maximum at mean position.
(D) The acceleration is minimum at extreme points.
Choose the correct answer from the options given below.

1 (C) and (D) only

2 (A), (C) and (D) only

3 (A), (B) and (C) only

4 (A), (B) and (D) only

Explanation:

In a simple harmonic motion, acceleration is proportional to the displacement of the particle measured from its mean position (equilibrium position) and always directed towards the mean position.
$∴ - \frac{d^{2} x}{d t^{2}} α x \Rightarrow a α (- x)$ .
Hence $\vec{a}$ and $\vec{x}$ are oppositely directed.
At the extreme position $x$ is maximum, hence ' $a$ ' is maximum.

JEE - 2023

PHXI14:OSCILLATIONS

364155 Assertion :
The soldiers marching on a suspended bridge are advised to go out of steps.
Reason :
Frequency of marching steps may match with the natural frequency of oscillation of bridge.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364156 Statement A :
An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Statement B :
The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI14:OSCILLATIONS

364157 Assertion :
The vibration of a polyatomic molecule about its equilibrium configuration is an example of SHM .
Reason :
Polyatomic molecule satisfy all conditions necessary for simple harmonic motion.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364153 A heavy small-sized sphere is suspended by a string of length $l$ . The sphere rotates uniformly in a horizontal circle with the string making an angle $θ$ with the vertical. Then, the time period of this conical pendulum is

1

t = 2 π \sqrt{\frac{l}{g}}

2

t = 2 π \sqrt{\frac{l \sin θ}{g}}

3

t = 2 π \sqrt{\frac{l \cos θ}{g}}

4

t = 2 π \sqrt{\frac{l}{g \cos θ}}

Explanation:

Radius of circular path in the horizontal plane
$Δ D A B, r = l \sin θ$
Resolving $T$ along the vertical and horizontal directions, we get
$\begin{matrix} (1) & T \cos θ = M g \end{matrix}$
supporting img
$T \sin θ = M r ω^{2} = M (l \sin θ) ω^{2}$
$\begin{matrix} (2) & \Rightarrow T = M l ω^{2} \end{matrix}$
On dividing Eq. (2) by Eq. (1), we get
$\frac{1}{\cos θ} = \frac{l ω^{2}}{g} \Rightarrow ω^{2} = \frac{g}{l \cos θ}$
$∴$ Time period, $t = \frac{2 π}{ω} = 2 π \sqrt{\frac{l \cos θ}{g}}$

PHXI14:OSCILLATIONS

364154 In a linear Simple Harmonic Motion ( $S H M$ )
(A) Restoring force is directly proportion to the displacement.
(B) The acceleration and displacement are opposite in direction.
(C) The velocity is maximum at mean position.
(D) The acceleration is minimum at extreme points.
Choose the correct answer from the options given below.

1 (C) and (D) only

2 (A), (C) and (D) only

3 (A), (B) and (C) only

4 (A), (B) and (D) only

Explanation:

In a simple harmonic motion, acceleration is proportional to the displacement of the particle measured from its mean position (equilibrium position) and always directed towards the mean position.
$∴ - \frac{d^{2} x}{d t^{2}} α x \Rightarrow a α (- x)$ .
Hence $\vec{a}$ and $\vec{x}$ are oppositely directed.
At the extreme position $x$ is maximum, hence ' $a$ ' is maximum.

JEE - 2023

PHXI14:OSCILLATIONS

364155 Assertion :
The soldiers marching on a suspended bridge are advised to go out of steps.
Reason :
Frequency of marching steps may match with the natural frequency of oscillation of bridge.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364156 Statement A :
An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Statement B :
The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI14:OSCILLATIONS

364157 Assertion :
The vibration of a polyatomic molecule about its equilibrium configuration is an example of SHM .
Reason :
Polyatomic molecule satisfy all conditions necessary for simple harmonic motion.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364153 A heavy small-sized sphere is suspended by a string of length $l$ . The sphere rotates uniformly in a horizontal circle with the string making an angle $θ$ with the vertical. Then, the time period of this conical pendulum is

1

t = 2 π \sqrt{\frac{l}{g}}

2

t = 2 π \sqrt{\frac{l \sin θ}{g}}

3

t = 2 π \sqrt{\frac{l \cos θ}{g}}

4

t = 2 π \sqrt{\frac{l}{g \cos θ}}

Explanation:

Radius of circular path in the horizontal plane
$Δ D A B, r = l \sin θ$
Resolving $T$ along the vertical and horizontal directions, we get
$\begin{matrix} (1) & T \cos θ = M g \end{matrix}$
supporting img
$T \sin θ = M r ω^{2} = M (l \sin θ) ω^{2}$
$\begin{matrix} (2) & \Rightarrow T = M l ω^{2} \end{matrix}$
On dividing Eq. (2) by Eq. (1), we get
$\frac{1}{\cos θ} = \frac{l ω^{2}}{g} \Rightarrow ω^{2} = \frac{g}{l \cos θ}$
$∴$ Time period, $t = \frac{2 π}{ω} = 2 π \sqrt{\frac{l \cos θ}{g}}$

PHXI14:OSCILLATIONS

364154 In a linear Simple Harmonic Motion ( $S H M$ )
(A) Restoring force is directly proportion to the displacement.
(B) The acceleration and displacement are opposite in direction.
(C) The velocity is maximum at mean position.
(D) The acceleration is minimum at extreme points.
Choose the correct answer from the options given below.

1 (C) and (D) only

2 (A), (C) and (D) only

3 (A), (B) and (C) only

4 (A), (B) and (D) only

Explanation:

In a simple harmonic motion, acceleration is proportional to the displacement of the particle measured from its mean position (equilibrium position) and always directed towards the mean position.
$∴ - \frac{d^{2} x}{d t^{2}} α x \Rightarrow a α (- x)$ .
Hence $\vec{a}$ and $\vec{x}$ are oppositely directed.
At the extreme position $x$ is maximum, hence ' $a$ ' is maximum.

JEE - 2023

PHXI14:OSCILLATIONS

364155 Assertion :
The soldiers marching on a suspended bridge are advised to go out of steps.
Reason :
Frequency of marching steps may match with the natural frequency of oscillation of bridge.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364156 Statement A :
An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Statement B :
The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI14:OSCILLATIONS

364157 Assertion :
The vibration of a polyatomic molecule about its equilibrium configuration is an example of SHM .
Reason :
Polyatomic molecule satisfy all conditions necessary for simple harmonic motion.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364153 A heavy small-sized sphere is suspended by a string of length $l$ . The sphere rotates uniformly in a horizontal circle with the string making an angle $θ$ with the vertical. Then, the time period of this conical pendulum is

1

t = 2 π \sqrt{\frac{l}{g}}

2

t = 2 π \sqrt{\frac{l \sin θ}{g}}

3

t = 2 π \sqrt{\frac{l \cos θ}{g}}

4

t = 2 π \sqrt{\frac{l}{g \cos θ}}

Explanation:

Radius of circular path in the horizontal plane
$Δ D A B, r = l \sin θ$
Resolving $T$ along the vertical and horizontal directions, we get
$\begin{matrix} (1) & T \cos θ = M g \end{matrix}$
supporting img
$T \sin θ = M r ω^{2} = M (l \sin θ) ω^{2}$
$\begin{matrix} (2) & \Rightarrow T = M l ω^{2} \end{matrix}$
On dividing Eq. (2) by Eq. (1), we get
$\frac{1}{\cos θ} = \frac{l ω^{2}}{g} \Rightarrow ω^{2} = \frac{g}{l \cos θ}$
$∴$ Time period, $t = \frac{2 π}{ω} = 2 π \sqrt{\frac{l \cos θ}{g}}$

PHXI14:OSCILLATIONS

364154 In a linear Simple Harmonic Motion ( $S H M$ )
(A) Restoring force is directly proportion to the displacement.
(B) The acceleration and displacement are opposite in direction.
(C) The velocity is maximum at mean position.
(D) The acceleration is minimum at extreme points.
Choose the correct answer from the options given below.

1 (C) and (D) only

2 (A), (C) and (D) only

3 (A), (B) and (C) only

4 (A), (B) and (D) only

Explanation:

In a simple harmonic motion, acceleration is proportional to the displacement of the particle measured from its mean position (equilibrium position) and always directed towards the mean position.
$∴ - \frac{d^{2} x}{d t^{2}} α x \Rightarrow a α (- x)$ .
Hence $\vec{a}$ and $\vec{x}$ are oppositely directed.
At the extreme position $x$ is maximum, hence ' $a$ ' is maximum.

JEE - 2023

PHXI14:OSCILLATIONS

364155 Assertion :
The soldiers marching on a suspended bridge are advised to go out of steps.
Reason :
Frequency of marching steps may match with the natural frequency of oscillation of bridge.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364156 Statement A :
An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Statement B :
The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI14:OSCILLATIONS

364157 Assertion :
The vibration of a polyatomic molecule about its equilibrium configuration is an example of SHM .
Reason :
Polyatomic molecule satisfy all conditions necessary for simple harmonic motion.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364153 A heavy small-sized sphere is suspended by a string of length $l$ . The sphere rotates uniformly in a horizontal circle with the string making an angle $θ$ with the vertical. Then, the time period of this conical pendulum is

1

t = 2 π \sqrt{\frac{l}{g}}

2

t = 2 π \sqrt{\frac{l \sin θ}{g}}

3

t = 2 π \sqrt{\frac{l \cos θ}{g}}

4

t = 2 π \sqrt{\frac{l}{g \cos θ}}

Explanation:

Radius of circular path in the horizontal plane
$Δ D A B, r = l \sin θ$
Resolving $T$ along the vertical and horizontal directions, we get
$\begin{matrix} (1) & T \cos θ = M g \end{matrix}$
supporting img
$T \sin θ = M r ω^{2} = M (l \sin θ) ω^{2}$
$\begin{matrix} (2) & \Rightarrow T = M l ω^{2} \end{matrix}$
On dividing Eq. (2) by Eq. (1), we get
$\frac{1}{\cos θ} = \frac{l ω^{2}}{g} \Rightarrow ω^{2} = \frac{g}{l \cos θ}$
$∴$ Time period, $t = \frac{2 π}{ω} = 2 π \sqrt{\frac{l \cos θ}{g}}$

PHXI14:OSCILLATIONS

364154 In a linear Simple Harmonic Motion ( $S H M$ )
(A) Restoring force is directly proportion to the displacement.
(B) The acceleration and displacement are opposite in direction.
(C) The velocity is maximum at mean position.
(D) The acceleration is minimum at extreme points.
Choose the correct answer from the options given below.

1 (C) and (D) only

2 (A), (C) and (D) only

3 (A), (B) and (C) only

4 (A), (B) and (D) only

Explanation:

In a simple harmonic motion, acceleration is proportional to the displacement of the particle measured from its mean position (equilibrium position) and always directed towards the mean position.
$∴ - \frac{d^{2} x}{d t^{2}} α x \Rightarrow a α (- x)$ .
Hence $\vec{a}$ and $\vec{x}$ are oppositely directed.
At the extreme position $x$ is maximum, hence ' $a$ ' is maximum.

JEE - 2023

PHXI14:OSCILLATIONS

364155 Assertion :
The soldiers marching on a suspended bridge are advised to go out of steps.
Reason :
Frequency of marching steps may match with the natural frequency of oscillation of bridge.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.

PHXI14:OSCILLATIONS

364156 Statement A :
An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Statement B :
The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1 Statement A is correct but Statement B is incorrect.

2 Statement A is incorrect but Statement B is correct.

3 Both statements are correct.

4 Both Statements are incorrect.

PHXI14:OSCILLATIONS

364157 Assertion :
The vibration of a polyatomic molecule about its equilibrium configuration is an example of SHM .
Reason :
Polyatomic molecule satisfy all conditions necessary for simple harmonic motion.

1 Both Assertion and Reason are correct and Reason is the correct explanation of the Assertion.

2 Both Assertion and Reason are correct but Reason is not the correct explanation of the Assertion.

3 Assertion is correct but Reason is incorrect.

4 Assertion is incorrect but reason is correct.