Grade 12Mechanics

Please give a detailed solution for this question...

Aishwarya

8 Years agoGrade 12

2 Answers

Arun

8 Years ago

We can find the effective mass of the spring by finding its kinetic energy. This requires adding all the length elements' kinetic energy, and requires the following integral:

$T =\int_m\frac{1}{2}u^2\,dm$

Since the spring is uniform, $dm=\left(\frac{dy}{L}\right)m$ , where $L$ is the length of the spring. Hence,

$=\frac{1}{2}\frac{m}{L}\int_0^L u^2\,dy$ $T = \int_0^L\frac{1}{2}u^2\left(\frac{dy}{L}\right)m\!$

The velocity of each mass element of the spring is directly proportional to its length, i.e. $u=\frac{vy}{L}$ , from which it follows:

$T =\frac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy$

$=\frac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy$

$=\frac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L$ hence the answer $=\frac{1}{2}\frac{m}{3}v^2$

Aishwarya

8 Years ago

How do you explain u=vy/l , can you please give some mathematical explanation or just some logic behind the same?

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Please give a detailed solution for this question...

Aishwarya

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Arun

Aishwarya

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