Answer:
[tex]\displaystyle w=3.478\ rad/sec[/tex]
[tex]M=0.0182\ J[/tex]
[tex]v=0.398\ m/s[/tex]
Explanation:
Simple Pendulum
It's a simple device constructed with a mass (bob) tied to the end of an inextensible rope of length L and let swing back and forth at small angles. The movement is referred to as Simple Harmonic Motion (SHM).
(a) The angular frequency of the motion is computed as
[tex]\displaystyle w=\sqrt{\frac{g}{L}}[/tex]
We have the length of the pendulum is L=0.81 meters, then we have
[tex]\displaystyle w=\sqrt{\frac{9.8}{0.81}}[/tex]
[tex]\displaystyle w=3.478\ rad/sec[/tex]
(b) The total mechanical energy is computed as the sum of the kinetic energy K and the potential energy U. At its highest point, the kinetic energy is zero, so the mechanical energy is pure potential energy, which is computed as
[tex]U=mgh[/tex]
where h is measured to the reference level (the lowest point). Please check the figure below, to see the desired height is denoted as Y. We know that
[tex]H+Y=L[/tex]
And
[tex]H=L\ cos\alpha[/tex]
Solving for Y
[tex]Y=L(1-cos\alpha )[/tex]
[tex]Since\ \alpha=8.1^o, L=0.81\ m[/tex]
[tex]Y=0.0081\ m[/tex]
The potential energy is
[tex]U=mgh=0.23\ kg(9.8\ m/s^2)(0.0081\ m)[/tex]
[tex]U=0.0182\ J[/tex]
The mechanical energy is, then
[tex]M=K+U=0+U=U[/tex]
[tex]M=0.0182\ J[/tex]
(c) The maximum speed is achieved when it passes through the lowest point (the reference for h=0), so the mechanical energy becomes all kinetic energy (K). We know
[tex]\displaystyle K=\frac{mv^2}{2}[/tex]
Equating to the mechanical energy of the system (M)
[tex]\displaystyle \frac{mv^2}{2}=0.0182[/tex]
Solving for v
[tex]\displaystyle v=\sqrt{\frac{(2)(0.0182)}{0.23}}[/tex]
[tex]v=0.398\ m/s[/tex]
