91Ó°ÊÓ

The term force may be defined as the product of mass and its acceleration.

Consider the given data as below.

The length of both strings L=50 cm

The mass of upper is $m_1=2kg$

The mass of lower ball is$m_2=3kg$

The initial height of the ball is $h=10cm$

The angular momentum of the simple pendulum with small amplitude is calculated as

$\mathrm{Ó\neg }=2\mathrm{Ï€}f$

$$\begin{gathered} f=\frac{1}{2\mathrm{π}}\sqrt{\frac{g}{L}} \\ =\frac{1}{2×3.14}\sqrt{\frac{9.8}{50×{10}^{−2}}} \\ =0.7\mathrm{Hz} \end{gathered}$$

The potential energy converted in kinetic energy at the point of maximum speed

$$\begin{gathered} \mathrm{mgh}=\frac{1}{2}{\mathrm{mv}}_{max}^2 \\ {\mathrm{v}}_{max}=\sqrt{2\mathrm{gh}} \\ {\mathrm{v}}_{max}=\sqrt{2×9.8×10×{10}^{−2}} \\ {\mathrm{v}}_{max}=1.4\mathrm{m}/\mathrm{s} \end{gathered}$$

Consider before collision the speed of mass$m_1$is${\mathrm{μ}}_1$and the speed of mass $m_2$is ${\mathrm{μ}}_2$and after collision the speed of mass $m_1$is $v_1$and the speed of mass $m_2$is $v_2$. So according to law of conservation of mass,

${\mathrm{m}}_1{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{u}}_2={\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2$

Before collision:

$$\begin{gathered} {\mathrm{u}}_1={\mathrm{v}}_{\max } \\ {\mathrm{v}}_{\max }=1.4\mathrm{m}/\mathrm{s} \\ {\mathrm{u}}_2=0\mathrm{m}/\mathrm{s} \end{gathered}$$

After collision:

$$\begin{gathered} v_1=v_2 \\ V=v_2 \end{gathered}$$

Than

$m_1{v}_{max}+m_2×0=(m_1+m_2)V$

$\begin{array}{r}V=\frac{m_1{v}_{max}}{m_1+m_2}\\ =\frac{2×1.4}{2+3}\\ =0.56\mathrm{m}/\mathrm{s}\end{array}$

Now the maximum displacement after collision the kinetic energy converted into potential energy so

$Mg{h}_1=\frac{1}{2}M{V}^2$

From $h_1$calculated as

$$\begin{gathered} {\mathrm{h}}_1=\frac{{\mathrm{V}}^2}{2\mathrm{g}} \\ =\frac{(0.56{)}^2}{2×9.8} \\ =0.016\mathrm{m} \\ =1.6\mathrm{cm} \end{gathered}$$

Hence, the maximum displacement after collision is

Now calculated the angular displacement,

$$\begin{gathered} cos(\mathrm{θ})=\frac{base}{hypotenuse} \\ =\frac{48.4}{50} \\ =co{s}^{−1}\left(\frac{48.4}{50}\right) \\ =0.254rad \\ cos\left(\mathrm{θ}\right)=0.254rad×\frac{{180}^{∘}}{\mathrm{π}rad} \\ =14{.53}^{∘} \end{gathered}$$

Hence, the frequency and maximum angular displacement of the motion after the collision is f =0.7 Hz and $\mathrm{θ}=0.254rad$respectively