Tag: shm as projection of circular motion

Questions Related to shm as projection of circular motion

The particle executes SHM on a straight line. At two positions its velocity $u$ and $v$ while acceleration, $\alpha$ and $\beta$ respectively $[\beta > \alpha >0]$, the distance between the two positions will be:-

physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

  1. $\frac{u^2+v^2}{\alpha+\beta}$

  2. $-\frac{u^2-v^2}{\alpha+\beta}$

  3. $\frac{u^2-v^2}{\alpha-\beta}$

  4. $\frac{u^2+v^2}{\beta-\alpha}$

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Correct Option: B
Explanation:
$u=w\sqrt{A^2-X _1^2}$
$V=w\sqrt{A^2-X _2^2}$
$\alpha=-w^2 x _1^2$
$\beta=-w^2 x _2^2$
$\frac{u^2-v^2}{u^2}=-(x _1^2-x _2^2)$
$=-(x _1+x _2)(x _1-x _2)$
$=\frac{(\alpha+\beta)}{w^2}(x _1-x _2)$
$x _1-x _2=\frac{u^2-v^2}{\alpha+\beta}$

A stone is projected from the ground with a velocity of $14 \ ms^{-1}$ one second later it clean a wall $2 \ m$ high. The angle of projection is $(g = 10 \ ms^{-2})$ 

physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

  1. $30 ^\circ$

  2. $45 ^\circ$

  3. $60 ^\circ$

  4. $15 ^\circ$

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Correct Option: A

The period of a particle it is $8s$. At $t=0$ it is at the mean position. The ratio of the distance covered by the particle in first second and second will be

physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

  1. $\cfrac { \sqrt { 2 } -1 }{ \sqrt { 2 } } $

  2. $\cfrac { 1 }{ \sqrt { 2 } } $

  3. $\cfrac { 1 }{ \sqrt { 2 } -1 } $

  4. $\left[ \sqrt { 2 } -1 \right] $

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Correct Option: C
Explanation:

$x=A\sin\omega t$

$T= 8 sec$
$\dfrac{2\pi}{\omega}=8 sec$
$\omega=\dfrac{2\pi}{8}=\dfrac{\pi}{4}$
$x=A\sin \dfrac{\pi}{4} t$ ,At $t=1 sec$
$x _1=\dfrac{A}{\sqrt2}$ ,At $t=2sec$
$x _2=A$
Required ration $=\dfrac{x _1}{x _2-x _1}=\dfrac{1/\sqrt2}{1-\dfrac{1}{\sqrt2}}=\dfrac{1}{\sqrt2-1}$

Sunlight of intensity 1.3 kW m–2 is incident normally on a thin convex lens of focal length 20 cm. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light, kW m–2, at a distance 22 cm from the lens on the other side is _________. 

physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

  1. $ 130 $

  2. $120 \ $

  3. $ 170$

  4. $160 $

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Correct Option: A,C

Simple harmonic motion is the projection of uniform circular motion on the

physics simple harmonic motion representing shm with circular motion shm as projection of circular motion simple harmonic motion (shm) as a projection of uniform circular motion

  1. $x$- axis

  2. $y$- axis

  3. reference circle

  4. any diameter of reference circle.

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Correct Option: D
Explanation:

Simple harmonic motion is the projection of uniform circular motion on any diameter of reference circle.