The speed of the hoop when it has rolled halfway up the side of the pipe is √(v₀² - gR).
Conservation of energy
The speed of the hoop when it has rolled halfway up the side of the pipe is calculated as follows;
K.E = P.E
- ¹/₂mv₀² + ¹/₂Iω² = (mgh₀ - mghf)
- ¹/₂mv₀² + ¹/₂Iω² = (0 - 0.5mgh) (hf = 0.5h) (half way up)
¹/₂Iω² = ¹/₂mv₀² - 0.5mgh
where;
- I is moment of inertia of the hoop = mr²
- ω is angular speed = v/r
¹/₂(mr²)(vf/r)² = ¹/₂mv₀² - 0.5mgh
¹/₂vf² = ¹/₂v₀² - ¹/₂gh
vf² = v₀² - gh
vf = √(v₀² - gh)
where;
- h is the distance traveled half-way up the pipe = R
vf = √(v₀² - gR)
[tex]v_f = \sqrt{v_0^2 - gR}[/tex]
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