Frequency is?

Classical Mechanics Level 4

A mass M M is in static equilibrium on a massless vertical spring as shown in the figure. A ball of mass m m dropped from certain height sticks to the mass M M after colliding with it. The oscillations they perform reach to height 'a' above the original level of scales and depth 'b' below it. Which of the following option is correct regarding the oscillation frequency

?

Independent of M and m Dependent on m Dependent on both M and m Dependent on M

Tanishq Varshney by Tanishq Varshney , 23, India

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2 solutions

Tanishq Varshney
Dec 29, 2015

where x o x_{o} is the distance of new mean position from the original position.

Clearly about the new mean position

a − x o = b + x o a-x_{o}=b+x_{o}

⇒ x o = a − b 2 \Rightarrow x_o =\frac{a-b}{2}

From F = k x o F=k x_{o}

⇒ ( M + m ) g = k x o ⇒ k = 2 ( M + m ) g a − b \Rightarrow (M+m)g=k x_o \Rightarrow k=\frac{2(M+m)g}{a-b}

f = 1 2 π k M + m \large{f=\frac{1}{2 \pi} \sqrt{\frac{k}{M+m}}}

⇒ f = 1 2 π 2 g ( a − b ) \large{\Rightarrow f=\frac{1}{2 \pi} \sqrt{\frac{2g}{(a-b)}}}

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The frequency is not independent of M and m. Actually both a and b are dependent on both the masses. Independent implies that changing M and m won't change the frequency.

Pranav Rao - 5 years, 5 months ago

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Can u provide an exact answer for the frequency?

Tanishq Varshney - 5 years, 5 months ago

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@Kishore S Shenoy

Tanishq Varshney - 5 years, 5 months ago

Actually, the question specifies a particular value of a a and b b . Thus the frequency depends on m m and M M

Kishore S. Shenoy - 5 years, 5 months ago

Actually, the mistake lies in the fact that the new equilibrium position is below the initial one... So a < b a<b

Kishore S. Shenoy - 5 years, 5 months ago
Anirban Mandal
Dec 30, 2015

1/2pi Root ( k/[m+M] ) is the answer ... which is obvious and even doesn't require any analysis ... where k is the spring constant of the spring

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Ofcourse ... the answer may be expressed in terms of a and b but I have to see how this can be done ... though it should be easy ... and my exams ar approaching ... so maybe later I will answer this

Anirban Mandal - 5 years, 5 months ago

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What if the ( m + M ) (m+M) term cancels after substitution?

Kishore S. Shenoy - 5 years, 5 months ago

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