Physics 4 Marks Question

1. $\ell=3\text{cm}=0.03\text{m}.$

$\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}}}=2\pi\sqrt{\frac{0.03}{9.8}}=0.34$ secound.

2. When the lady sets on the Merry-go-round the ear rings also experience centrepetal acceleration,

$\text{a}=\frac{\text{v}^2}{\text{r}}=\frac{\text{4}^2}{2}=8\text{m/s}^2$

Resultant Acceleration $\text{A}=\sqrt{\text{g}^2+\text{a}^2}=\sqrt{100+64}=12.8\text{m/s}^2$

Time period $\text{T}=2\pi\sqrt{\frac{\ell}{\text{A}}}=2\pi\sqrt{\frac{0.03}{12.8}}=0.30$ second.

From the freebody diagram,

$\text{T}=\sqrt{(\text{mg})^2+\Big(\frac{\text{mv}^2}{\text{r}^2}\Big)}$

$=\text{m}\sqrt{\text{g}^2+\frac{\text{v}^4}{\text{r}^2}}=\text{ma}$ where a = acceleration $=\Big(\text{g}^2+\frac{\text{v}^4}{\text{r}^2}\Big)^{\frac{1}{2}}$

The time period of small accellations is given by,

$\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}}}=2\pi\sqrt{\frac{\ell}{\Big(\text{g}^2+\frac{\text{v}^4}{\text{r}^2}\Big)^{\frac{1}{2}}}}$

$\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}}}$

$\Rightarrow4=2\pi\sqrt{\frac{\ell}{\text{g}}}\ ...(1)$

$\text{T}=2\pi\sqrt{\frac{\ell}{\text{g}^2+\text{a}_0^2}}$

$\Rightarrow3.99=2\pi\sqrt{\frac{\ell}{\text{g}^2+\text{a}_0^2}}$