Answer:
The equation of the system after the collision
[tex]\alpha(t)=17.5*sin(3.13t)[/tex]
Explanation:
The motion is an inelastic collision so, the mass o after the collision are a new mass of both mass so the first step is use the conservation of linear momentum
[tex]m_{1}*v_{1}=(m_{2}+m_{1})*v_{f}[/tex]
solve to vf
[tex]v_{f}=\frac{m_{1}*v_{1}}{m_{2}+m_{1}}=\frac{0.1kg*20\frac{m}{s}}{2.0kg+0.1kg}=0.952 \frac{m}{s}[/tex]
The conservation of mechanical energy so:
[tex]E_{1}=E_{2}[/tex]
[tex]\frac{1}{2}*(m_{1}+m{2})*v^2=(m_{1}+m_{2})*g*h[/tex]
[tex]h=l*[1-cos(\alpha_{max})][/tex]
[tex]\frac{1}{2}*v^2=g*l[1-cos(\alpha_{max})][/tex]
solve to α
[tex]\alpha_{max}=cos^{-1}*[1-\frac{v_{f}^2}{2*g*l}][/tex]
[tex]\alpha_{max}=cos^{-1}*[1-\frac{0.952^2}{2*9.8*1}][/tex]
[tex]\alpha_{max}=17.5[/tex]
Finally angular frequency when the motion become as a pendulum
[tex]w=\sqrt{\frac{g}{l}}=\sqrt{\frac{9.8}{1}}[/tex]
[tex]w=3.13 \frac{rad}{s}[/tex]
The equation of the motion of the system after the collision is
[tex]\alpha(t)=\alpha_{max}*sin(wt)[/tex]
[tex]\alpha(t)=17.5*sin(3.13t)[/tex]
