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ItsImpulse
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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?
Starting with zero speed?ItsImpulse said:suppose you had a ball rolling down a ramp, without slipping
Thrown horizontally from same height as the first ball?ItsImpulse said:and compare it to a ball that starts with a velocity u that is horizontally to the side.
Consider the vertical accelerations in both cases.ItsImpulse said:how would the time taken be different to reach the bottom?
A.T. said:Starting with zero speed?
Thrown horizontally from same height as the first ball?
Consider the vertical accelerations in both cases.
CWatters said: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.
For sliding. Rotational inertia makes it even slower.ItsImpulse said:3. vertical acceleration is just gsin(theta) am I right?
ItsImpulse said: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?
CWatters said: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.
The angle of the ramp does affect the time it takes for a ball to roll down. The steeper the angle, the faster the ball will roll. This is due to the force of gravity pulling the ball down at a greater rate. As the angle decreases, the time it takes for the ball to roll down also increases.
Yes, the material of the ramp can impact the time difference for a ball to roll down. A smoother material, such as a metal ramp, will provide less friction and allow the ball to roll faster. Rougher materials, such as sandpaper, will create more friction and slow down the ball's descent.
The mass of the ball does not have a significant impact on the time difference for it to roll down a ramp. As long as the mass is consistent, the gravitational force will act on the ball at the same rate, regardless of its mass. However, a heavier ball may have more momentum and continue to roll for a longer distance after reaching the bottom of the ramp.
Yes, there is a formula that can be used to calculate the time difference for a ball to roll down a ramp. It is t = √(2h/g), where t is the time in seconds, h is the height of the ramp in meters, and g is the acceleration due to gravity (9.8 m/s²). Keep in mind that this formula assumes a frictionless surface and a ball starting from rest at the top of the ramp.
Air resistance can have a small impact on the time difference for a ball to roll down a ramp. If the ball is rolling at high speeds, the air resistance may slightly slow it down, causing a slightly longer time difference. However, for most experiments with a ramp and a small ball, the effect of air resistance can be considered negligible.