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physics ncert solutions class 11th Physics Class 11th Oscillations

14.7 The motion of a particle executing simple harmonic motion is described by the displacement function,

x(t) = A cos ( $ω$ t + $φ$ ).

If the initial (t = 0) position of the particle is 1 cm and its initial velocity is $ω$ cm/s, what are its amplitude and initial phase angle ? The angular frequency of the particle is $π$ s^–1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin ( $ω$ t + $α$ ), what are the amplitude and initial phase of the particle with the above initial conditions.

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Vishal Baghel | Contributor-Level 10

5 months ago

Initially at t =0, Displacement, x = 1 cm

Initial velocity, v = $ω$ cm/s, Angular frequency, $ω$ = $π$ rad/s

It is given that: x(t) = A cos ( $ω$ t + $φ$ )

Then 1 = A cos ( $ω \times 0 + φ)$

A cos $φ$ = 1 ….(i)

Velocity, v = $\frac{d x}{d t}$

$ω = - A ω s i n ? (ω t + φ)$

1 = -Asin( $ω \times 0 + φ) = - A s i n φ$

Asin $φ = - 1$ ………(ii)

Squaring and adding equations (i) and (ii), we get

$A^{2} ({s i n}^{2} φ + {c o s}^{2} φ)$ = 1+1

$A^{2} = 2$

A = $\sqrt 2$ cm

$\tan ? φ$ = -1

Then $φ$ = $\frac{3 π}{4}$ , $\frac{7 π}{4}$ ,,,

SHM is given as X = Bsin( $ω$ t + $α$ )

Putting the given values in this equation, we get

1 = Bsin( $ω \times 0 + α$ )

Bsin $α$ =1…….(iii)

Velocity V = $ω$

...more

Initially at t =0, Displacement, x = 1 cm

Initial velocity, v = $ω$ cm/s, Angular frequency, $ω$ = $π$ rad/s

It is given that: x(t) = A cos ( $ω$ t + $φ$ )

Then 1 = A cos ( $ω \times 0 + φ)$

A cos $φ$ = 1 ….(i)

Velocity, v = $\frac{d x}{d t}$

$ω = - A ω s i n ? (ω t + φ)$

1 = -Asin( $ω \times 0 + φ) = - A s i n φ$

Asin $φ = - 1$ ………(ii)

Squaring and adding equations (i) and (ii), we get

$A^{2} ({s i n}^{2} φ + {c o s}^{2} φ)$ = 1+1

$A^{2} = 2$

A = $\sqrt 2$ cm

$\tan ? φ$ = -1

Then $φ$ = $\frac{3 π}{4}$ , $\frac{7 π}{4}$ ,,,

SHM is given as X = Bsin( $ω$ t + $α$ )

Putting the given values in this equation, we get

1 = Bsin( $ω \times 0 + α$ )

Bsin $α$ =1…….(iii)

Velocity V = $ω$ Bcos( $ω$ t + $α$ )

Substituting the given values, we get $π$ = $π$ Bcos $α$ or B $\cos ? α$ = 1 ….(iv)

Squaring and adding equations (iii) and (iv), we get

$B^{2} ({s i n}^{2} α + {c o s}^{2} α)$ = 1+1

B = $\sqrt{2} c m$

Dividing equation (iii) by equation (iv), we get

$\frac{B \sin ? α}{B \cos ? α}$ = 1 or $\tan ? α$ = 1

$α$ = $\frac{π}{4}$ , $\frac{5 π}{4}$

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