A thin uniform rod, of length 30d and mass m, is bent to form a frame - Leaving Cert Applied Maths - Question 7 - 2021
Question 7
A thin uniform rod, of length 30d and mass m, is bent to form a frame. The frame is in the shape of a right-angled triangle ABC, as shown in the diagram.
$|AC| = 13... show full transcript
Worked Solution & Example Answer:A thin uniform rod, of length 30d and mass m, is bent to form a frame - Leaving Cert Applied Maths - Question 7 - 2021
Step 1
Find $|BC|$ in terms of d
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Answer
To find ∣BC∣, we can use the Pythagorean theorem, given that triangle ABC is a right triangle. The relationship can be established as follows:
Let ∣BC∣=x. Then:
∣AC∣2=∣AB∣2+∣BC∣2
Substituting the known lengths:
(13d)2=(17d−x)2+x2
Expanding this equation leads us to:
169d2=(17d−x)2+x2
From this equation, we can find that:
∣BC∣=5d
Step 2
Find the distance of the centre of gravity of the frame from AB
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Answer
To find the distance from AB, we first calculate the moments about point A. The mass distribution can be expressed as:
m_{AB} = rac{m}{30} imes |AB|^2 ext{ and } m_{AC} = rac{m}{30} imes |AC|^2
Utilizing these, we set up the equilibrium condition:
m_{AC} rac{5d}{2} + m_{BC} rac{5d}{2} = m imes y
Substituting the values leads us to find
y = rac{2}{3}d
Step 3
Find the value of k
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Answer
To find the value of k, we analyze the forces acting on point A:
kmg imes 12d = mg imes rac{2}{3}d
Rearranging yields:
k = rac{1}{8}
Step 4
Find T, the tension in the string, in terms of W, $\ell$ and h
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Answer
Using the triangle formed by rod PQ and the wall, we analyze the vertical and horizontal components. The equilibrium equation is:
Timesextsineθ=Wimes21∣PQ∣=W×21∣PQ∣sinα
Thus,
T=2hWℓ.
Step 5
If $T = \frac{1}{3}W$, find $\ell$ in terms of h
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Answer
Setting T=31W into the previous equation gives us:
31W=2hWℓ
After simplifying, we find:
ℓ=32h
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