Question

Take a spring of medium stiffness, about 20 cm in length. Fix the upper end of the spring to a rigid support. On the lower end attach a scale pan of known mass so that the system hangs vertically. Pull the pan 1-2 cm downwards and release. The pan will oscillate up and down. Measure the time of 20 oscillations with the help of a stop watch and calculate the time period of oscillation of pan. Now place a middle size pebble on the pan, set it in oscillations and determine the time period of oscillation again. Calculate the mass of the pebble using the data obtained from your observations.

Answer 1

So what is your question?

Answer 2

"Here M is Mass of the pan and pebble, so mass of the pan ‘m’ needs to be subtracted from M for the final mass.

$T_1 = 2 \pi \sqrt{\frac {M} {K}}\quad \rightarrow(1)$

$T_{ 2 }=2\pi \sqrt { \frac { (m+M) }{ K } }\quad \rightarrow(2)$

Squaring 1 and 2,

$T_{ 1 }=\frac { 4\pi ^{ 2 }M }{ K }\quad \rightarrow(3)$

$T_2 = \frac { 4\pi ^{ 2 }(m+M) }{ K }\quad \rightarrow (4)$

Dividing 3 by 4

$\frac { (T_{ 1 }^{ 2 }) }{ (T_{ 2 }^{ 2 }) } =\frac { m }{ (m+M) }$

$T_{ 2 }^{ 2 }=T_{ 1 }^{ 2 }+(1+\frac { M }{ m } )$

$T_{ 2 }^{ 2 }-T_{ 1 }^{ 2 }=\frac { (MT_{ 1 }^{ 2 }) }{ m }$

$M =\frac{m(T^2_2-T^2_1)}{T^2_1}$"