reference:

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=150.msg6487#msg6487 by Hwang Fu Kwun original

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1801.0 by Hwang Fu Kwun original layout by Ahmed.

**Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop.**This work is licensed under a Creative Commons Attribution 2.5 Taiwan License

- Please feel free to post your ideas about how to use the simulation for better teaching and learning.
- Post questions to be asked to help students to think, to explore.
- Upload worksheets as attached files to share with more users.

Description:

Slipping and Rolling Sphere

taken and adapted to sphere from http://www.compadre.org/osp/items/detail.cfm?ID=8606

The EJS Slipping and Rolling Sphere Model shows the motion of sphere rolling on a floor subject to a frictional force as determined by the coefficient of friction μk. The simulation allows the user to change the initial translational and rotational velocities of the wheel, v_i and ω_i, and the radius, mass and mass distribution momentum of inertia cofficeient, R, m, and C of the wheel. By controlling these variables, the dynamics of the wheel can be changed to show the slipping, then rolling without slipping, of the wheel.

Slipping and Rolling Object (Sphere or Cylindrical) Theory

taken and adapted to sphere from http://www.compadre.org/osp/items/detail.cfm?ID=8606

The theory behind the simulation of the rolling and slipping sphere is relatively straightforward, but substantially differs from its simpler cousin: the rolling without slipping sphere. When a sphere rolls without slipping, v = ωR, where v is the linear or translational velocity of the wheel, ω is the angular or rotational velocity of the wheel, and R is the wheel’s radius. When this condition is maintained, the relative velocity between the bottom of the wheel and the ground is zero. This condition, therefore, also means that the frictional force that acts must be static friction which cannot do any work on the wheel, and energy methods can be used to analyze the motion.

For rolling with slipping, the force acting on the wheel is kinetic friction which must be treated both as a force, F, acting on the center of mass of the wheel and as a torque, τ, acting at the point of contact between the wheel and the ground. During this motion, both the force and the torque are constant and therefore the velocity and the angular velocity can be determined with the constant acceleration kinematics equations: v = v0 +(F/m)t and ω = ω0 +(τ/I)t, respectively The rotational and translational motions are then independent until v = ωR when the wheel begins to roll without slipping. The time when this occurs can be found with these kinematics equations solving for when v = ωR.

The simulation has these theoretical details explicitly encoded in it. Specifically:

The translation and rotational motions each have their own differential equation (ODE) describing the motion (dx/dt = v, dv/dt = F/m, dθ/dt = ω, and dω/dt = τ/I).

These two motions can be coupled in the same way that two-dimensional motion in x and y can be coupled, but the dynamics can be understood separately.

EJS makes it easy to model the coupled problem without messy mathematical manipulations with a lot of trigonometric functions. The Evolution page in EJS allows one to easily transform from the space to the body frame by changing the transform vectors. See the Evolution workpanel in EJS for details.

EJS differential equation solver (ODE) events allow us to determine precisely when slipping stops, v = ωR, and to switch the equations of motion from rolling with slipping (constant acceleration and constant angular acceleration) to rolling without slipping (constant velocity and constant angular velocity). See the ODE events page in EJS for details.

Exercises:

taken and adapted to sphere from http://www.compadre.org/osp/items/detail.cfm?ID=8606

Questions

1. For an initially translating, but not rotating, wheel, draw a free-body diagram and determine the acceleration and the angular acceleration of the wheel for the time it is translating and slipping. Your answers should be written in terms of μk, v_i, C kw, m, and R.

2. For the scenario in (1), (a) determine the time the wheel is slipping, (b) determine the distance the wheel travels while slipping, (c) determine the final translational and rotational velocities when the slipping stops. Check your answer against the simulation.

3. Redo questions (1) and (2) for an initially rotating (with backspin), but not translating, wheel. Check your answer against the simulation.

4. Redo questions (1) and (2) for an initially rotating (with backspin), and initially translating, wheel. Check your answer against the simulation. Show that the condition for the wheel to end up stationary is that v_i = -v*ω_i*R*kw.