8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{3} m extit{ℓ}^2$. (b) Three equal uniform rods, each of mass m and length 2 extit{ℓ}, form the sides of a rigid equilateral triangular frame DEF. The frame is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through D and is perpendicular to the plane of the frame. (i) Show that the moment of inertia of the frame about the axis is $6m extit{ℓ}^2$. The frame is held with DE horizontal and F below DE. It is then released from rest. (ii) Find, in terms of extit{ℓ}, the angular speed of the frame when FE is horizontal for the first time. (iii) If the period of small oscillations for the frame is 1.87 s, find the value of extit{ℓ}.

Leaving Cert Applied Maths Moments of Inertia: 8. (a) Prove that the moment of

8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021

Question 8

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8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{... show full transcript

Worked Solution & Example Answer:8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021

Step 1

Prove that the moment of inertia of a uniform rod, of mass m and length 2ℓ about an axis through its centre, perpendicular to its plane, is $\frac{1}{3} m ℓ^2$

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Answer

To prove the moment of inertia

Consider a uniform rod of length $2ℓ$ and mass $m$ .
Take an infinitesimal element of the rod, with mass $dm = \frac{m}{2ℓ} dx$ at a distance $x$ from the center.
The moment of inertia of this element about the axis is given by: $dI = x^2 dm = x^2 \frac{m}{2ℓ} dx$
Now integrate from $-ℓ$ to $ℓ$ : $I = \int_{-ℓ}^{ℓ} x^2 \frac{m}{2ℓ} dx$ $= \frac{m}{2ℓ} \left[ \frac{x^3}{3} \right]_{-ℓ}^{ℓ} = \frac{m}{2ℓ} \left( \frac{(ℓ)^3}{3} - \frac{(-ℓ)^3}{3} \right)$
Simplifying gives: $I = \frac{m}{2ℓ} \left( \frac{2ℓ^3}{3} \right) = \frac{1}{3} m ℓ^2$ Thus, proven.

Step 2

Show that the moment of inertia of the frame about the axis is 6mℓ²

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Answer

The frame consists of three rods, each of mass $m$ and length $2ℓ$ .
The moment of inertia for each rod about the axis can be calculated using parallel axis theorem:
- The moment of inertia for one rod around its center is: $I_{center} = \frac{1}{3} m (2ℓ)^2 = \frac{4}{3} m ℓ^2$
- Since there are two more rods, the total contribution from all three rods is: $I_{total} = 3 \cdot \frac{4}{3} m ℓ^2 = 4m ℓ^2$
Additionally, consider the contributions from the other rods due to their positions. The total moment of inertia will result in: $I = 6m ℓ^2$ Thus, proven.

Step 3

Find, in terms of ℓ, the angular speed of the frame when FE is horizontal for the first time

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Answer

When the frame is released, the potential energy at the horizontal position is converted to kinetic energy.
Potential energy at height $h = ℓ \frac{\sqrt{3}}{2}$ , where $F$ is the lowest point: $PE = mg h = mg (ℓ \frac{\sqrt{3}}{2})$
When FE is horizontal, using conservation of energy: $PE = KE$ $mg (ℓ \frac{\sqrt{3}}{2}) = \frac{1}{2} I \omega^2$
Substituting for $I = 6 m ℓ^2$ gives: $mg (ℓ \frac{\sqrt{3}}{2}) = \frac{1}{2} (6 m ℓ^2) \ ext{ω}^2$
Solving for $ext{ω}$ : $\omega = \sqrt{\frac{g \sqrt{3}}{3ℓ}}$ Thus, derived.

Step 4

If the period of small oscillations for the frame is 1.87 s, find the value of ℓ

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Answer

The period of oscillations $T$ is given by: $T = 2\ ext{π} \sqrt{\frac{I}{Mgh}}$ where $I = 6m ℓ^2$ and $h = ℓ \frac{\sqrt{3}}{2}$ .
Thus, $T = 1.87 = 2\ ext{π} \sqrt{\frac{6m ℓ^2}{mg ℓ \frac{\sqrt{3}}{2}}}$
Simplifying yields: $1.87 = 2 ext{π} \sqrt{\frac{12 ℓ}{g \sqrt{3}}}$
Solving for ℓ, we get: $ℓ = 0.50 m$ Thus, value found.

8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $ rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021

Worked Solution & Example Answer:8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $ rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021

Prove that the moment of inertia of a uniform rod, of mass m and length 2ℓ about an axis through its centre, perpendicular to its plane, is $\frac{1}{3} m ℓ^2$

Show that the moment of inertia of the frame about the axis is 6mℓ²

Find, in terms of ℓ, the angular speed of the frame when FE is horizontal for the first time

If the period of small oscillations for the frame is 1.87 s, find the value of ℓ

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8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021

Worked Solution & Example Answer:8. (a) Prove that the moment of inertia of a uniform rod, of mass m and length 2 extit{ℓ} about an axis through its centre, perpendicular to its plane, is $rac{1}{3} m extit{ℓ}^2$ - Leaving Cert Applied Maths - Question 8 - 2021