Max Distance Up Ramp for Rotating Sphere

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The discussion revolves around calculating the maximum distance a uniform solid sphere travels up a ramp at an angle of 27.6° after rolling without sliding. In the first scenario, where the ramp is frictionless, the initial kinetic energy is incorrectly equated, leading to an erroneous conclusion about the distance. For the second scenario, with sufficient friction preventing sliding, the calculations confirm that the approach is correct, allowing for both linear and rotational motion to stop instantaneously. The correct formula for the distance in the first case is derived as l = v²/(2g*sinθ), clarifying the misunderstanding. Overall, the key takeaway is the distinction between the two scenarios and the importance of accurately accounting for energy types.
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A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 2.30 m/s when it encounters a ramp that is at an angle θ = 27.6° above the horizontal. Find the maximum distance that the sphere travels up the ramp if :
1-
the ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.
→ I used : Ki+Ui=Kf+Uf
and concluded that Ki=7/10* m*v² =mglsinθ
so l = (7/10 *m*v²)/(mg*sinθ).

is it true ?
2-
the ramp has enough friction to prevent the sphere from sliding so that both the linear and rotational motion stop (instantaneously).
→ I concluded the same as the first case : means l = (7/10 *m*v²)/(mg*sinθ).
is it correct ?
 
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physicos said:
A uniform solid sphere of mass M and radius R is rolling without sliding along a level plane with a speed v = 2.30 m/s when it encounters a ramp that is at an angle θ = 27.6° above the horizontal. Find the maximum distance that the sphere travels up the ramp if :
1-
the ramp is frictionless, so the sphere continues to rotate with its initial angular speed until it reaches its maximum height.
→ I used : Ki+Ui=Kf+Uf
and concluded that Ki=7/10* m*v² =mglsinθ
so l = (7/10 *m*v²)/(mg*sinθ).

is it true ?
Nope.

2-
the ramp has enough friction to prevent the sphere from sliding so that both the linear and rotational motion stop (instantaneously).
→ I concluded the same as the first case : means l = (7/10 *m*v²)/(mg*sinθ).
is it correct ?
It turns out this one is correct. Obviously the two cases are different, so you need to figure out where you messed up in the first part of the problem.
 
Where did the 7/10 come from?

Does the sphere have rotational kinetic energy?
 
The sphere has both rotational and transitional kinetic energy
 
vela said:
Nope.It turns out this one is correct. Obviously the two cases are different, so you need to figure out where you messed up in the first part of the problem.

FOR THE FIRST CASE :

Kf+Uf=Ki+Ui
so 1/2m*v²f+1/2*I*w²+mg*l*sinθ=1/2*m*v²+1/2*I*w²
so It becomes : mg*l*sinθ=1/2*m*v²so l = v²/2*g*sinθ

Is it correct now ?
 
That part looks right to me.
 

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