A solid ball rolls perfectly with initial velocity v0 in horizontal axis ( y-axis ) on an inclined plane with elevation angle Θ as the picture above shown. This ball moves turning due to the gravitational acceleration till it has traveled distance L in x axis when it's at the bottom of the plane.
Determine the time (t) the ball needs to get to the bottom of the plane ! (The ball doesn't slip while rolling)
Rotational dynamics equation and linear kinematics equation
Or conservation of energy equation
The Attempt at a Solution
I have two methods to solve the problem. But, the answers are different.
Using rot. dynamics equation and linear kinematics.
I just consider the x-axis since it's what the question asks.
ΣFx = ma
mg sin Θ - f = ma (Note : f is friction force)
f = mg sin Θ - ma
Στ = I α
f R = I α
f R = I (a/R)
(mg sin Θ - ma) R = (2/5) m R^2 (a/R)
mg sin Θ - ma = (2/5) m a
g sin Θ - a = (2/5) a
(7/5) a = g sin Θ
a = (5/7) g sin Θ
Then, I use the kinematics equation
L = 0.5 a t^2
2L/a = t^2
14L/ (5g sin Θ) = t^2
t = √(14L/5g sinΘ)
But, using conservation of energy, I get different answer
m g sin Θ L = (1/2) m v^2 + (1/2) I ω^2
m g sin Θ L = (1/2) m v^2 + (1/2) (2/5 m R^2) (v^2 / R^2)
g sin Θ L = (1/2) v^2 + (1/5) m v^2
g sin Θ L = (7/10) v^2
v = √(10 g sin Θ L / 7 )
Then, I use the kinematics
vt = vox + a t
√(10 g sin Θ L / 7 ) = 0 + g sin Θ t
t = √(10L/7g sin Θ)
Which one is correct? Why?
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