Ball rolling down a ramp time difference

[ItsImpulse]

suppose you had a ball rolling down a ramp, without slipping and compare it to a ball that starts with a velocity u that is horizontally to the side. how would the time taken be different to reach the bottom?

 

[A.T.]

suppose you had a ball rolling down a ramp, without slipping

Starting with zero speed?

and compare it to a ball that starts with a velocity u that is horizontally to the side.

Thrown horizontally from same height as the first ball?

how would the time taken be different to reach the bottom?

Consider the vertical accelerations in both cases.

 

[ItsImpulse]

Starting with zero speed?


Thrown horizontally from same height as the first ball?


Consider the vertical accelerations in both cases.

1. yes starting with 0 speed.
2. it's rolling down a ramp but yes at same height.
3. vertical acceleration is just gsin(theta) am I right?

 

[CWatters]

Perhaps look at it from an energy perspective. Both start with PE but one is rolling and the other not. Apply conservation of energy. They can't both have the same linear KE at the bottom. The one that's just falling/sliding will have converted all of the initial PE to linear KE. The one that's rolling will have converted some to rotational KE leaving less for linear KE.

 

[ItsImpulse]

Perhaps look at it from an energy perspective. Both start with PE but one is rolling and the other not. Apply conservation of energy. They can't both have the same linear KE at the bottom. The one that's just falling/sliding will have converted all of the initial PE to linear KE. The one that's rolling will have converted some to rotational KE leaving less for linear KE.

so in other words the one that rotates more will go down the ramp slower?

it would be mgh = 0.5mv^2 + 0.5Iw^2 right?

 

[A.T.]

3. vertical acceleration is just gsin(theta) am I right?

For sliding. Rotational inertia makes it even slower.

 

[CWatters]

so in other words the one that rotates more will go down the ramp slower?

it would be mgh = 0.5mv^2 + 0.5Iw^2 right?

Correct.

Whereas for a block or ball sliding down a frictionless inclined surface it's just mgh = 0.5mv^2.

So the final velocity must be different.

Aside: In both cases we're ignoring energy losses to friction but there must be some friction in the case of the ball that's rolling or it wouldn't start rotating.

 

[Delta2]

In the case of the rolling (without sliding) ball, friction doesn't do work and there arent energy loses. The pseudo-work of friction (equal to Friction X length of ramp) equals the final rotational kinetic energy of the ball.

 

[jbriggs444]

Correct.

Whereas for a block or ball sliding down a frictionless inclined surface it's just mgh = 0.5mv^2.

So the final velocity must be different.

Aside: In both cases we're ignoring energy losses to friction but there must be some friction in the case of the ball that's rolling or it wouldn't start rotating.

That component of friction is accounted for. Hence the 0.5 I ω² term. Rolling resistance, if any, is not accounted for.

 

[CWatters]

Yes sorry. It was the rolling resistance I meant was being ignored.

 

[olivermsun]

I think the OP is asking whether an additional component of motion in the plane of the ramp (at right angles to both "downslope" and "normal") would change the time it takes for the ball to reach the bottom.