JEE MAIN - Physics (2019 - 8th April Evening Slot - No. 7)
In a simple pendulum experiment for
determination of acceleration due to gravity (g),
time taken for 20 oscillations is measured by
using a watch of 1 second least count. The
mean value of time taken comes out to be
30 s. The length of pendulum is measured by
using a meter scale of least count 1 mm and the
value obtained is 55.0 cm. The percentage
error in the determination of g is close to :-
0.2%
3.5%
0.7%
6.8%
Explanation
Time period of a pendulum (T) = $$2\pi \sqrt {{l \over g}} $$
$$ \Rightarrow $$ T2 = $$4{\pi ^2}{l \over g}$$
$$ \Rightarrow $$ $$g = {{4{\pi ^2}l} \over {{T^2}}}$$
Fractional change
$$\left( {{{dg} \over g}} \right) \times 100 = \left( {{{dl} \over l}} \right) \times 100 - \left( {2{{dT} \over T}} \right) \times 100$$
$$ \therefore $$ Maximum possible percentage error,
$$\left( {{{dg} \over g}} \right) \times 100 = \left( {{{dl} \over l}} \right) \times 100 + \left( {2{{dT} \over T}} \right) \times 100$$
Error in time period(dT) = least count of time = 1 second
and T = 30 second
Error in length(dl) = least count of length = 1 mm
and $$l$$ = 55.0 cm
$$ \therefore $$ $$\left( {{{dg} \over g}} \right) \times 100 =$$ $$\left( {{{0.1} \over {55}}} \right) \times 100 + 2\left( {{1 \over {30}}} \right) \times 100$$ = 6.8%
$$ \Rightarrow $$ T2 = $$4{\pi ^2}{l \over g}$$
$$ \Rightarrow $$ $$g = {{4{\pi ^2}l} \over {{T^2}}}$$
Fractional change
$$\left( {{{dg} \over g}} \right) \times 100 = \left( {{{dl} \over l}} \right) \times 100 - \left( {2{{dT} \over T}} \right) \times 100$$
$$ \therefore $$ Maximum possible percentage error,
$$\left( {{{dg} \over g}} \right) \times 100 = \left( {{{dl} \over l}} \right) \times 100 + \left( {2{{dT} \over T}} \right) \times 100$$
Error in time period(dT) = least count of time = 1 second
and T = 30 second
Error in length(dl) = least count of length = 1 mm
and $$l$$ = 55.0 cm
$$ \therefore $$ $$\left( {{{dg} \over g}} \right) \times 100 =$$ $$\left( {{{0.1} \over {55}}} \right) \times 100 + 2\left( {{1 \over {30}}} \right) \times 100$$ = 6.8%
Comments (0)
