Name: Condition for Resonance
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: Do you know you can shatter the glass goblet without touching it? Yes, it is possible when the sound of a specific frequency is produced. Let's find out the...

Question

A thin uniform plate shaped as an equilateral triangle with a height

h

performs small oscillations about the horizontal axis coinciding with one of its sides. Find the oscillation period and the reduced length of the given pendulum.

Accepted Answer

To apply the energy method for obtaining the angular frequency for this physical pendulum, we need the moment of inertia of the triangular plate about the axis $A B$ .

Consider a small element $d x$ at a depth $x$ from the axis $A B$ for this element we can write,

$I = \int x^{2} d m = \int x^{2} ρ d x d y = \int_{0}^{h} x^{2} ρ d x \frac{h - x}{h} \cdot \frac{2 h}{\sqrt{3}} = \int_{0}^{h} x^{2} \frac{2 ρ}{\sqrt{3}} (h - x) d x = \frac{2 ρ}{\sqrt{3}} (\frac{h^{4}}{3} - \frac{h^{4}}{4}) = \frac{ρ h^{4}}{6 \sqrt{3}} = \frac{m h^{2}}{6}$

We know that the area of the triangle $Δ A B C = \frac{h^{2}}{\sqrt{3}}$ and $m = ρ Δ$ .

Therefore $K E = \frac{1}{2} \frac{m h^{2}}{6} ω^{2}$

$P E = m g \frac{h}{3} (1 - \cos θ) = m g \frac{h}{3} [2 \sin^{2} (\frac{θ}{2})] \approx m g \frac{h}{3} [2 {(\frac{θ}{2})}^{2}] \Rightarrow P E = \frac{1}{2} m g h \frac{θ^{2}}{3}$ ,

where $θ$ is the instantaneous angle that the plane of the triangular plate with the absolute vertical, and $ω$ is the angular velocity at this instant.

Here, the total energy of the system at any instant is,
$E = \frac{1}{2} \frac{m h^{2}}{6} ω^{2} + \frac{1}{2} \frac{m g h}{3} θ^{2} = constant$ .

Differentiating we get,

$\frac{1}{2} \frac{m h^{2}}{6} 2 ω \cdot \frac{d ω}{d t} + \frac{1}{2} \frac{m g h}{3} 2 θ \frac{d θ}{d t} = 0$ .

Solving we get,

$\frac{d^{2} θ}{d t^{2}} = - (\frac{2 g}{h}) θ$ .

Therefore, $angular {frequency}^{2} = \frac{2 g}{h} = \frac{\frac{m g h}{3}}{\frac{m h^{2}}{6}}$ .
So, $T = 2 π \sqrt{\frac{h}{2 g}} = π \sqrt{\frac{2 h}{g}}$ and

Comparing with the time period of simple pendulum, $T = 2 π \sqrt{\frac{l}{g}}$ . $l_{reduced} = \frac{h}{2}$ .

A thin uniform plate shaped as an equilateral triangle with a height $h$ performs small oscillations about the horizontal axis coinciding with one of its sides. Find the oscillation period and the reduced length of the given pendulum.

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