Q5. two springs of constants k1 and k2 have equal maximum velocities w

1.	Q5. two springs of constants k1 and k2 have equal maximum velocities whenexecuting SHM. The ratio of their amplitudes will be (masses are equal)-
Answer» Question : Two SPRINGS of constants k1 and k2 have equal maximum VELOCITIES when executing SHM,the ratio of their amplitudes will be (MASSES are equal) : Solution : The springs have equal maximum velocities and masses Their springs constants are $\sf K_1 \ \sf{and} \ K_2$ Maximum Velocity is GIVEN by : $\sf \: \: v {}_{max} = A \omega$ Also, $\sf \: \omega \: = \sqrt{ \dfrac{k}{m} }$ THUS, $\sf \: v = {A}_{1} \times \sqrt{ \dfrac{ K_{1} }{M} } - - - - - - - (1)$ Also, $\sf \: v = A_{2} \times \sqrt{ \dfrac{ K_{2} }{M} } - - - - - - -(2)$ Equating equations (1) and (2),as maximum velocities of the SHM are equal $\longrightarrow \: \sf \: A_{1} \times \sqrt{ \dfrac{ K_{1} }{M} } = A_{2} \times \sqrt{ \dfrac{ {K}_{2} }{M} } \\ \\ \longrightarrow \: \sf{ \dfrac{ A_{1} }{ A_{2} } = \sqrt{ \dfrac{ K_{2} }{ K_{1} } } } \\ \\ \longrightarrow \: \boxed{ \boxed{\sf{ \dfrac{ A_{1} }{ A_{2} } = (\frac{ K_{1} }{ {K}_{2} } ) {}^{ \frac{1}{2} } }}}$

Q5. two springs of constants k1 and k2 have equal maximum velocities whenexecuting SHM. The ratio of their amplitudes will be (masses are equal)-

Answer»

Question :

Two SPRINGS of constants k1 and k2 have equal maximum VELOCITIES when executing SHM,the ratio of their amplitudes will be (MASSES are equal) :

Solution :

The springs have equal maximum velocities and masses

Their springs constants are $\sf K_1 \ \sf{and} \ K_2$

Maximum Velocity is GIVEN by :

$\sf \: \: v {}_{max} = A \omega$

Also,

$\sf \: \omega \: = \sqrt{ \dfrac{k}{m} }$

THUS,

$\sf \: v = {A}_{1} \times \sqrt{ \dfrac{ K_{1} }{M} } - - - - - - - (1)$

Also,

$\sf \: v = A_{2} \times \sqrt{ \dfrac{ K_{2} }{M} } - - - - - - -(2)$

Equating equations (1) and (2),as maximum velocities of the SHM are equal

$\longrightarrow \: \sf \: A_{1} \times \sqrt{ \dfrac{ K_{1} }{M} } = A_{2} \times \sqrt{ \dfrac{ {K}_{2} }{M} } \\ \\ \longrightarrow \: \sf{ \dfrac{ A_{1} }{ A_{2} } = \sqrt{ \dfrac{ K_{2} }{ K_{1} } } } \\ \\ \longrightarrow \: \boxed{ \boxed{\sf{ \dfrac{ A_{1} }{ A_{2} } = (\frac{ K_{1} }{ {K}_{2} } ) {}^{ \frac{1}{2} } }}}$