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Homework Statement
A particle is at rest at the apex A of a smooth fixed hemisphere whose base is horizontal. The hemisphere has centre O and radius a. The particle is then displaced very slightly from rest and moves on the surface of the hemisphere. At the point P on the surface where angle AOP = α the particle has speed v. Find an expression for v in terms of a, g and α.
Homework Equations
The Attempt at a Solution
So I’ve worked like this:
Total energy at A = PE + KE = amg + 0
Total energy at P = PE + KE = (0.5m(v^2)) + xmg
x = a – y
(cos α)/y = (sin90)/a => y = a(cos α)
=> x = a – (a(cos α)) = a (1  cos α)
=> Total energy at P = PE + KE = (0.5m(v^2)) + amg (1  cos α)
Therefore (Total energy at A) = (Total energy at P) gives
amg = (0.5m(v^2)) + amg (1  cos α)
ag = 0.5(v^2) + ag (1 – cos α)
ag – ag (1 – cos α) = 0.5(v^2)
2ag (1 – 1 + cos α) = v^2
v = sqrt (2ag (cos α))
However, the correct answer is v = sqrt (2ag (1  cos α))
Where’s the problem?
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