Solving an Inertial Mystery: Angular Acceleration and Mud

[PhysicsRock]

Homework Statement

A wheel of radius $R$ is spinning at angular velocity $\omega$. What distance does a pedestrian walking behind the vehicle have to walk, to not be hit by a piece of mud falling off the wheel.

Relevant Equations

None really, it's all about the thought.

So, my idea would be that this happens at an angle $\theta = \frac{\pi}{2}$, or quarter of a whole rotation. At this point, the wheel starts moving right again, after going to the left. Due to it's inertia, the piece of mud would want to keep it's current direction of motion and therefore fall off. The problem I'm facing now is that the acceleration always points to the center of the wheel, thus the mud would be accelerated towards the cars movement direction, rather than being thrown backward where it could hit the person.
Is my idea of the angle wrong or did I calculate the acceleration vector incorrectly? Help is very apprechiated.

 

[jbriggs444]

Homework Statement:: A wheel of radius $R$ is spinning at angular velocity $\omega$. What distance does a pedestrian walking behind the vehicle have to walk, to not be hit by a piece of mud falling off the wheel.
Relevant Equations:: None really, it's all about the thought.

So, my idea would be that this happens at an angle $\theta = \frac{\pi}{2}$, or quarter of a whole rotation. At this point, the wheel starts moving right again, after going to the left. Due to it's inertia, the piece of mud would want to keep it's current direction of motion and therefore fall off. The problem I'm facing now is that the acceleration always points to the center of the wheel, thus the mud would be accelerated towards the cars movement direction, rather than being thrown backward where it could hit the person.
Is my idea of the angle wrong or did I calculate the acceleration vector incorrectly? Help is very apprechiated.

I would suggest that mud or rain flies off of a spinning wheel at every possible rotation angle. It takes a variable length of time for a drop of mud to form and then drop free from the tire. Certainly, the seat of my pants has stood in mute testament to the fact that it sometimes takes more than a quarter of a rotation for the rear wheel in my bicycle to fling road grime up and forward.

Your observation that the acceleration of the mud is always inward is a good one. The rotating wheel cannot ever fling any material rearward. No portion of the rotating wheel is ever moving rearward relative to the ground.

The pedestrian can be hit by splashing mud. Or by mud flung forward by a rotating wheel.

 

[PhysicsRock]

I would suggest that mud or rain flies off of a spinning wheel at every possible rotation angle. It takes a variable length of time for a drop of mud to form and then drop free from the tire. Certainly, the seat of my pants has stood in mute testament to the fact that it sometimes takes more than a quarter of a rotation for the rear wheel in my bicycle to fling road grime up and forward.

Your observation that the acceleration of the mud is always inward is a good one. The rotating wheel cannot ever fling any material rearward. No portion of the rotating wheel is ever moving rearward relative to the ground.

The pedestrian can be hit by splashing mud. Or by mud flung forward by a rotating wheel.

Thank you so much! The last paragraph just gave me the idea I needed. Your answer was super helpful. :)

 

[kuruman]

The rotating wheel cannot ever fling any material rearward. No portion of the rotating wheel is ever moving rearward relative to the ground.

I think "ever" is too absolute. Mud is notoriously slippery. If the wheel slips as it translates forward so that $V<\omega R$, there is a good chance that mud can be flung rearward from ground level.

 

[jbriggs444]

I think "ever" is too absolute. Mud is notoriously slippery. If the wheel slips as it translates forward so that $V<\omega R$, there is a good chance that mud can be flung rearward from ground level.

Yeah, a slipping wheel can do the trick. I was assuming rolling without slipping.

 

[haruspex]

Another complication is that the pedestrian is walking. How fast? Mud thrown in the first quadrant of rise will have less forward speed than the bicycle, so could hit a pedestrian moving almost as fast as the rider.

 

[kuruman]

When I was in college, I bought a cheap bike that didn't have a mudguard (fender?) over the rear wheel. The first time I rode it in wet weather, I ended up with a one-inch wide streak of filthy water running down the middle of the back of my shirt. That's when I consciously realized that the water that was flung from the top of the wheel traveled, relative to the ground, at about twice the speed of the back of my shirt.

 

[jrmichler]

Back when my daily driver was a car that I had bought for $100.00, some friends challenged my statement that driving skill was more important than four wheel drive when driving off road. I agreed to show them, provided that they wash the car afterward. The left front fender was missing.

I observed that the amount of mud flying off the front tire varied with speed. At low speed, the mud stuck to the tire. At "high" speed, it came off right away and very little was flung up into the air. At a certain speed, a large amount was flung up into the air, where it came down and covered the car. I think that speed was near either 15 or 25 MPH.

They had to spend some money and time at the car wash, but they did not complain.