Respuesta :
Answer:
0.087 m
Explanation:
Length of the rod, L = 1.5 m
Let the mass of the rod is m and d is the distance between the pivot point and the centre of mass.
time period, T = 3 s
the formula for the time period of the pendulum is given by
[tex]T = 2\pi \sqrt{\frac{I}{mgd}}[/tex] .... (1)
where, I is the moment of inertia of the rod about the pivot point and g is the acceleration due to gravity.
Moment of inertia of the rod about the centre of mass, Ic = mL²/12
By using the parallel axis theorem, the moment of inertia of the rod about the pivot is
I = Ic + md²
[tex]I = \frac{mL^{2}}{12}+ md^{2}[/tex]
Substituting the values in equation (1)
[tex]3 = 2 \pi \sqrt{\frac{\frac{mL^{2}}{12}+ md^{2}}{mgd}}[/tex]
[tex]9=4\pi^{2}\times \left ( \frac{\frac{L^{2}}{12}+d^{2}}{gd} \right )[/tex]
12d² -26.84 d + 2.25 = 0
[tex]d=\frac{26.84\pm \sqrt{26.84^{2}-4\times 12\times 2.25}}{24}[/tex]
[tex]d=\frac{26.84\pm 24.75}{24}[/tex]
d = 2.15 m , 0.087 m
d cannot be more than L/2, so the value of d is 0.087 m.
Thus, the distance between the pivot and the centre of mass of the rod is 0.087 m.
The pivot should be located at a distance of 0.736 meters from the center of the uniform rod.
Let suppose that the rod is a simple pendulum, whose period ([tex]T[/tex]), in seconds, is described by the following formula:
[tex]T = 2\pi\cdot \sqrt{\frac{l}{g} }[/tex] (1)
Where:
- [tex]l[/tex] - Pendulum length, in meters.
- [tex]g[/tex] - Gravitational acceleration, in meters per square second.
If we know that [tex]T = 3\,s[/tex] and [tex]g = 9.807\,\frac{m}{s^{2}}[/tex], then the pendulum length is:
[tex]\frac{T^{2}}{4\pi^{2}} = \frac{l}{g}[/tex]
[tex]l = \frac{T^{2}\cdot g}{4\pi^{2}}[/tex]
[tex]l = \frac{(3\,s)^{2}\cdot \left(9.807\,\frac{m}{s^{2}} \right)}{4\pi^{2}}[/tex]
[tex]l \approx 2.236\,m[/tex]
And the distance between the center of the uniform rod and the pivot is found by the following subtraction:
[tex]d = 2.236\,m - 1.50\,m[/tex]
[tex]d = 0.736\,m[/tex]
The pivot should be located at a distance of 0.736 meters from the center of the uniform rod.
To learn more on pendulums, we kindly invite to check this verified question: https://brainly.com/question/14759840
