What Is the Minimum Velocity Needed for a Wheel to Climb a Step?

[TyloBabe]

Homework Statement

A solid wheel of mass M and radius R rolls without slipping
at a constant velocity v until it collides inelastically with a step of height h < R.
Assume that there is no slipping at the point of impact. What is the minimum
velocity v in terms of h and R needed for the wheel to climb the step?

Homework Equations

Not sure. Perhaps conservation of Angular Momentum. We can not use energy because the collision is inelastic.

The Attempt at a Solution

I really have no clue. I know that Angular Momentum is conserved because the only real external force acting on the system is when the disc encounters the corner of the step, but that force always acts radially and so there is no torque.

I can only think that maybe the angular momentum about the discs radius, as well as about the point h, is conserved throughout the entire collision. Where do I begin?

 

[tiny-tim]

Hi TyloBabe!

Perhaps conservation of Angular Momentum. We can not use energy because the collision is inelastic.

Yes and no …

only the collision is inelastic, so you will need to use conservation of https://www.physicsforums.com/library.php?do=view_item&itemid=313" to find the angular velocity immediately after the collision …

but from then on you can assume that energy is conserved.

I really have no clue. I know that Angular Momentum is conserved because the only real external force acting on the system is when the disc encounters the corner of the step, but that force always acts radially and so there is no torque.

No, there is a torque, because the friction is tangential to the disc.

Since you don't know how much the friction is, the place to take moments about is … ?

 

[TyloBabe]

What I was thinking was saying that the initial angular momentum about the center of the disc, would be equal to the angular momentum of the disc about the point of contact on the step. Would that be correct?

I see what you mean about the torque. The normal force on the disc at the step acts radially, but the frictional force from the step acts tangentially in the opposite direction of motion.

Then if I can figure out what that velocity is just after the collision, and then use conservation of energy, i would just have to say that the rotational energy about the step would be equal to the rotational, translational, and gravitational energy just after the disc has made it over the step. Correct me if I'm mistaken, please.

 

[tiny-tim]

Hi TyloBabe!

What I was thinking was saying that the initial angular momentum about the center of the disc, would be equal to the angular momentum of the disc about the point of contact on the step. Would that be correct?

No, you can't change the point about which you measure the https://www.physicsforums.com/library.php?do=view_item&itemid=313"

Choose a point and stick to it!

In this case (btw, you didn't answer my question) that point would be the edge of the step … because there's no unknown https://www.physicsforums.com/library.php?do=view_item&itemid=175" about the edge (the friction has no torque there)

(Don't forget to include the initial translational angular momentum!)

I see what you mean about the torque. The normal force on the disc at the step acts radially, but the frictional force from the step acts tangentially in the opposite direction of motion.

Then if I can figure out what that velocity is just after the collision, and then use conservation of energy, i would just have to say that the rotational energy about the step would be equal to the rotational, translational, and gravitational energy just after the disc has made it over the step. Correct me if I'm mistaken, please.

Sort of … don't forget that you only need the final angular velocity to be zero!

 

[rwisz]

I would first start by identifying the key principles and equations that are relevant to this problem. In this case, the conservation of angular momentum and the lack of slipping at the point of impact are important factors to consider.

Next, I would consider the initial and final states of the system. Initially, the disc is rolling at a constant velocity v and has a certain amount of angular momentum. After the collision, the disc must have enough velocity to climb the step, which means it must have enough angular momentum to overcome the height of the step.

To find the minimum velocity v needed, we can use the conservation of angular momentum equation: L=Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. We can assume that the moment of inertia of the disc does not change during the collision, so we can equate the initial and final angular momenta.

Linitial = Lfinal
MvR = (MvR^2/2) + (Mv(h-R))

Solving for v, we get:

v = (h-R)/(1+R/2)

This is the minimum velocity needed for the disc to climb the step without slipping. Any velocity greater than this will also allow the disc to successfully climb the step.

In conclusion, by considering the principles of conservation of angular momentum and the lack of slipping at the point of impact, we can determine the minimum velocity needed for a disc to climb a step of height h < R.