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## Homework Statement

http://www.sumoware.com/images/temp/xzdmdjtlksnfhiqc.png [Broken]

A solid ball rolls perfectly with initial velocity v

_{0}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)

## Homework Equations

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.

ΣF

_{x}= 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

v

_{t}= v

_{ox}+ 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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