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PHXI14:OSCILLATIONS

364202 A particle executes $S . H . M .$ of amplitude $A$ along $x$ -axis. At $t = 0$ , the position of the particle is $x = \frac{A}{2}$ and it moves along positive $x$ -axis. The displacement of particle in time $t$ is $x = A \sin (ω t + δ)$ , then the value $δ$ will be

1

\frac{π}{4}

2

\frac{π}{6}

3

\frac{π}{2}

4

\frac{π}{3}

Explanation:

Equation of $S . H . M .$ is $x = A \sin (ω t + δ)$
supporting img
Phases diagram
At $t = 0, x = A \sin δ$ and $x = \frac{A}{2}$
So, $A / 2 = A \sin δ \Rightarrow \sin δ = 1 / 2$
$δ = 30^{\circ} = \frac{π}{6}$ .

JEE - 2023

PHXI14:OSCILLATIONS

364203 A simple harmonic motion having an amplitude $A$ and time period $T$ is represented by the equation :
$y = 5 \sin 2 π (t + 4 x) m$
Then, the values of $A$ (in $m$ ) and $T$ (in sec) are :

1

A = 5; T = 2

2

A = 10; T = 1

3

A = 5; T = 1

4

A = 10; T = 2

Explanation:

$A = 5, T = \frac{2 π}{2 π} = 1 s$

PHXI14:OSCILLATIONS

364204 $x = a \cos^{2} ω t$ or $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ represents
(i) Periodic motion but not simple harmonic motion if $x$ is displacement
(ii) SHM of angular frequency $2 ω$

1 Only (i) is correct

2 Only (ii) is correct

3 Both are incorrect

4 Both are correct

Explanation:

If $x$ is position, $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ . Here displacement $x$ varies simple harmonically with angular frequency $2 ω$ .

PHXI14:OSCILLATIONS

364205 Which one of the following equations of motion represents simple harmonic motion?

1 Acceleration

= - k_{0} x + k_{1} x^{2}

2 Acceleration

= - k (x + a)

3 Acceleration

= k (x + a)

4 Acceleration

= k x

Explanation:

Acceleration $\propto -$ (displacement).
$a \propto - y \Rightarrow a = - ω^{2} y$
$a = - \frac{k}{m} y Here, y = x + a a n d m = 1$
$∴ Acceleration = - k (x + a)$

PHXI14:OSCILLATIONS

364202 A particle executes $S . H . M .$ of amplitude $A$ along $x$ -axis. At $t = 0$ , the position of the particle is $x = \frac{A}{2}$ and it moves along positive $x$ -axis. The displacement of particle in time $t$ is $x = A \sin (ω t + δ)$ , then the value $δ$ will be

1

\frac{π}{4}

2

\frac{π}{6}

3

\frac{π}{2}

4

\frac{π}{3}

Explanation:

Equation of $S . H . M .$ is $x = A \sin (ω t + δ)$
supporting img
Phases diagram
At $t = 0, x = A \sin δ$ and $x = \frac{A}{2}$
So, $A / 2 = A \sin δ \Rightarrow \sin δ = 1 / 2$
$δ = 30^{\circ} = \frac{π}{6}$ .

JEE - 2023

PHXI14:OSCILLATIONS

364203 A simple harmonic motion having an amplitude $A$ and time period $T$ is represented by the equation :
$y = 5 \sin 2 π (t + 4 x) m$
Then, the values of $A$ (in $m$ ) and $T$ (in sec) are :

1

A = 5; T = 2

2

A = 10; T = 1

3

A = 5; T = 1

4

A = 10; T = 2

Explanation:

$A = 5, T = \frac{2 π}{2 π} = 1 s$

PHXI14:OSCILLATIONS

364204 $x = a \cos^{2} ω t$ or $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ represents
(i) Periodic motion but not simple harmonic motion if $x$ is displacement
(ii) SHM of angular frequency $2 ω$

1 Only (i) is correct

2 Only (ii) is correct

3 Both are incorrect

4 Both are correct

Explanation:

If $x$ is position, $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ . Here displacement $x$ varies simple harmonically with angular frequency $2 ω$ .

PHXI14:OSCILLATIONS

364205 Which one of the following equations of motion represents simple harmonic motion?

1 Acceleration

= - k_{0} x + k_{1} x^{2}

2 Acceleration

= - k (x + a)

3 Acceleration

= k (x + a)

4 Acceleration

= k x

Explanation:

Acceleration $\propto -$ (displacement).
$a \propto - y \Rightarrow a = - ω^{2} y$
$a = - \frac{k}{m} y Here, y = x + a a n d m = 1$
$∴ Acceleration = - k (x + a)$

PHXI14:OSCILLATIONS

364202 A particle executes $S . H . M .$ of amplitude $A$ along $x$ -axis. At $t = 0$ , the position of the particle is $x = \frac{A}{2}$ and it moves along positive $x$ -axis. The displacement of particle in time $t$ is $x = A \sin (ω t + δ)$ , then the value $δ$ will be

1

\frac{π}{4}

2

\frac{π}{6}

3

\frac{π}{2}

4

\frac{π}{3}

Explanation:

Equation of $S . H . M .$ is $x = A \sin (ω t + δ)$
supporting img
Phases diagram
At $t = 0, x = A \sin δ$ and $x = \frac{A}{2}$
So, $A / 2 = A \sin δ \Rightarrow \sin δ = 1 / 2$
$δ = 30^{\circ} = \frac{π}{6}$ .

JEE - 2023

PHXI14:OSCILLATIONS

364203 A simple harmonic motion having an amplitude $A$ and time period $T$ is represented by the equation :
$y = 5 \sin 2 π (t + 4 x) m$
Then, the values of $A$ (in $m$ ) and $T$ (in sec) are :

1

A = 5; T = 2

2

A = 10; T = 1

3

A = 5; T = 1

4

A = 10; T = 2

Explanation:

$A = 5, T = \frac{2 π}{2 π} = 1 s$

PHXI14:OSCILLATIONS

364204 $x = a \cos^{2} ω t$ or $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ represents
(i) Periodic motion but not simple harmonic motion if $x$ is displacement
(ii) SHM of angular frequency $2 ω$

1 Only (i) is correct

2 Only (ii) is correct

3 Both are incorrect

4 Both are correct

Explanation:

If $x$ is position, $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ . Here displacement $x$ varies simple harmonically with angular frequency $2 ω$ .

PHXI14:OSCILLATIONS

364205 Which one of the following equations of motion represents simple harmonic motion?

1 Acceleration

= - k_{0} x + k_{1} x^{2}

2 Acceleration

= - k (x + a)

3 Acceleration

= k (x + a)

4 Acceleration

= k x

Explanation:

Acceleration $\propto -$ (displacement).
$a \propto - y \Rightarrow a = - ω^{2} y$
$a = - \frac{k}{m} y Here, y = x + a a n d m = 1$
$∴ Acceleration = - k (x + a)$

PHXI14:OSCILLATIONS

364202 A particle executes $S . H . M .$ of amplitude $A$ along $x$ -axis. At $t = 0$ , the position of the particle is $x = \frac{A}{2}$ and it moves along positive $x$ -axis. The displacement of particle in time $t$ is $x = A \sin (ω t + δ)$ , then the value $δ$ will be

1

\frac{π}{4}

2

\frac{π}{6}

3

\frac{π}{2}

4

\frac{π}{3}

Explanation:

Equation of $S . H . M .$ is $x = A \sin (ω t + δ)$
supporting img
Phases diagram
At $t = 0, x = A \sin δ$ and $x = \frac{A}{2}$
So, $A / 2 = A \sin δ \Rightarrow \sin δ = 1 / 2$
$δ = 30^{\circ} = \frac{π}{6}$ .

JEE - 2023

PHXI14:OSCILLATIONS

364203 A simple harmonic motion having an amplitude $A$ and time period $T$ is represented by the equation :
$y = 5 \sin 2 π (t + 4 x) m$
Then, the values of $A$ (in $m$ ) and $T$ (in sec) are :

1

A = 5; T = 2

2

A = 10; T = 1

3

A = 5; T = 1

4

A = 10; T = 2

Explanation:

$A = 5, T = \frac{2 π}{2 π} = 1 s$

PHXI14:OSCILLATIONS

364204 $x = a \cos^{2} ω t$ or $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ represents
(i) Periodic motion but not simple harmonic motion if $x$ is displacement
(ii) SHM of angular frequency $2 ω$

1 Only (i) is correct

2 Only (ii) is correct

3 Both are incorrect

4 Both are correct

Explanation:

If $x$ is position, $x = \frac{a}{2} - \frac{a}{2} \cos 2 ω t$ . Here displacement $x$ varies simple harmonically with angular frequency $2 ω$ .

PHXI14:OSCILLATIONS

364205 Which one of the following equations of motion represents simple harmonic motion?

1 Acceleration

= - k_{0} x + k_{1} x^{2}

2 Acceleration

= - k (x + a)

3 Acceleration

= k (x + a)

4 Acceleration

= k x

Explanation:

Acceleration $\propto -$ (displacement).
$a \propto - y \Rightarrow a = - ω^{2} y$
$a = - \frac{k}{m} y Here, y = x + a a n d m = 1$
$∴ Acceleration = - k (x + a)$

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