In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. New amplitude of oscillation will be:

Option 1 -

$A \sqrt[]{\frac{M}{M - m}}$

Option 2 -

$A \sqrt[]{\frac{M}{M + m}}$

Option 3 -

$A \sqrt[]{\frac{M - m}{M}}$

Option 4 -

$A \sqrt[]{\frac{M + m}{M}}$

3 Views | Posted a month ago

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1 Answer

Answered by

alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

Velocity of block in equilibrium, in first case,

$v = A ω = A . \sqrt[]{\frac{k}{M}}$

Velocity of block in equilibrium, is second case,

$v' = A' ω' = A' \sqrt[]{\frac{k}{M + m}}$

From conservation of momentum,

Mv = (M + m) v’

$\Rightarrow M A \sqrt[]{\frac{k}{M}} = (M + m) A' \sqrt[]{\frac{k}{M + m}} \Rightarrow A' = A \sqrt[]{\frac{M}{M + m}}$

Velocity of block in equilibrium, in first case, <math> <mrow> <mi>v</mi> <mo>=</mo> <mi>A</mi> <mi>ω</mi> <mo>=</mo> <mi>A</mi> <mo>.</mo> <mroot> <mrow> <mfrac> <mrow> <mi>k</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1818922543"> </o:OLEObject></xml><! [endif]-> Velocity of block in equilibrium, is second case, <math> <mrow> <mi>v</mi> <mo>'</mo> <mo>=</mo> <mi>A</mi> <mo>'</mo> <mi>ω</mi> <mo>'</mo> <mo>=</mo> <mi>A</mi> <mo>'</mo> <mroot> <mrow> <mfrac> <mrow> <mi>k</mi> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1818922544"> </o:OLEObject></xml><! [endif]->   From conservation of momentum, Mv = (M + m) v’ <math> <mrow> <mo>⇒</mo> <mi>M</mi> <mi>A</mi> <mroot> <mrow> <mfrac> <mrow> <mi>k</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mrow> <mo> (</mo> <mrow> <mi>M</mi> <mo>+</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mi>A</mi> <mo>'</mo> <mroot> <mrow> <mfrac> <mrow> <mi>k</mi> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> <mo>⇒</mo> <mi>A</mi> <mo>'</mo> <mo>=</mo> <mi>A</mi> <mroot> <mrow> <mfrac> <mrow> <mi>M</mi> </mrow> <mrow> <mi>M</mi> <mo>+</mo> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow></mrow> </mroot> </mrow> </math>        <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1818922545"> </o:OLEObject></xml><! [endif]->

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