The rope will remain taut until the particle makes 79⁰ angle.
Change in kinetic energy of the particle
The change in kinetic energy of the particle is calculated as follows;
ΔK.E = K.Ei - K.Ef
Before the particle will achieve the given angular displacement, it will touch two new corners. Total kinetic energy lost = 30%
ΔK.E = 100%K.E - 30%K.E = 70%K.E = 0.7K.E
- let the vertical displacement of the particle = h
- horizontal length = side of the prism = a
- hypotenuse side = length of the pendulum = L
Apply principle of conservation of energy
K.E = P.E
0.7K.E = mgh
0.7(¹/₂mv²) = mg(Lsinθ)
0.7(v²) = 2g(Lsinθ)
from third kinematic equation;
v² = u² + 2gh
v² = 0 + 2gh
v² = 2g(a tanθ)
0.7(2g(a tanθ)) = 2g(Lsinθ)
0.7(a tanθ) = Lsinθ
0.7a/L = sinθ/tanθ
0.7a/L = cosθ
(0.7 x 0.8)/(3) = cosθ
0.1867 = cosθ
θ = cos⁻¹(0.1867)
θ = 79⁰
Thus, the rope will remain taut until the particle makes 79⁰ angle.
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