Answer:
Mass does not affect oscillation frequency.
Explanation:
Let the bob of the pendulum makes a small angular displacement θ. When the pendulum is displaced from the equilibrium position, a restoring force tries to act upon it and it tries to bring the pendulum back to its equilibrium position. Let this restoring force be F.
Therefore, F = -mgsinθ
Now for pendulum, for small angle of θ,
sinθ[tex]\simeq[/tex]θ
Therefore, F = -mgθ
Now from Newton's 2nd law of motion,
F = m.a = -mgθ
[tex]\Rightarrow m.\frac{d^{2}x}{dt^{2}} = - mg\Theta[/tex]
Now since, x = θ.L
[tex]\Rightarrow L.\frac{d^{2}\Theta }{dt^{2}}= -g\Theta[/tex]
[tex]\Rightarrow \frac{d^{2}\Theta }{dt^{2}}= -\frac{g}{L}.\Theta[/tex]
[tex]\Rightarrow \frac{d^{2}\Theta }{dt^{2}}+\frac{g}{L}.\Theta =0[/tex]
Therefore, angular frequency
[tex]\omega ^{2}[/tex] = [tex]\frac{g}{L}[/tex]
ω = [tex]\sqrt{\frac{g}{L}}[/tex]
Also we know angular frequency is , ω = 2.π.f
where f is frequency
Therefore
2πf = [tex]\sqrt{\frac{g}{L}}[/tex]
f = [tex]\frac{1}{2 \pi }\sqrt{\frac{g}{L}}[/tex]
So from here we can see that frequency,f is independent of mass, hence it does not affect frequency.
