Ball Rolling without slipping in a hill

[Fu-Kwun Hwang]

A ball or cylinder rolling (without slipping) down in a down hill slope.
The condition for rolling without slipping is
 v=R*ω (angular velocity) or a=R*α (angular acceleration)
where R is the radius of the ball or cylinder.

Assume the friction force is f
1. m*g*sinθ-f=m*a
2. R*f=I*α
3. I=(1/2)m*R² for cylinder, (2/5)m*R² for sphere
solve the above equation will give you
f=(1/3)m*g*sinθ for cylinder or f=(2/7)m*g*sinθ for sphere.

Embed a running copy of this simulation
<applet code='users.ntnu.fkh.ballNramp_pkg.ballNrampApplet.class' archive='http://www.phy.ntnu.edu.tw/ntnujava/ejsuser/2/ejs_ballNramp.jar' name='ballNramp3758' id='ballNramp3758' width='697' height='509' mayscript=true><param name='draggable' value='true'></applet>
Embed a running copy link(show simulation in a popuped window)
<a href='http://www.phy.ntnu.edu.tw/ntnujava/msg.php?smfejs=3758' target='smfejs'>Run simulation</a>

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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

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[Fu-Kwun Hwang]

The following add a spring make it more complicated!

You are welcomed to post your calculation for further discussion.

Embed a running copy of this simulation
<applet code='users.ntnu.fkh.ballNrampwspring_pkg.ballNrampwspringApplet.class' archive='http://www.phy.ntnu.edu.tw/ntnujava/ejsuser/2/ejs_ballNrampwspring.jar' name='ballNrampwspring3760' id='ballNrampwspring3760' width='697' height='509' mayscript=true><param name='draggable' value='true'></applet>
Embed a running copy link(show simulation in a popuped window)
<a href='http://www.phy.ntnu.edu.tw/ntnujava/msg.php?smfejs=3760' target='smfejs'>Run simulation</a>

Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list
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.

Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!

[lookang]

A ball or cylinder rolling (without slipping) down in a down hill slope.
The condition for rolling without slipping is
 v=R*ω (angular velocity) or a=R*α (angular acceleration)
where R is the radius of the ball or cylinder.

Assume the friction force is f
1. m*g*sinθ-f=m*a
2. R*f=I*α
3. I=(1/2)m*R² for cylinder, (2/5)m*R² for sphere
solve the above equation will give you
f=(1/3)m*g*sinθ for cylinder or f=(2/7)m*g*sinθ for sphere.

1. m*g*sinθ-f=m*a
2. R*f=I*α imply R*f = (1/2*m*R^2)*(a/R) imply 2*f/m = a

sub back

m*g*sinθ - f  = m*(2f/m) imply 1/3*m*g*sinθ = f for cylinder

[Fu-Kwun Hwang]

The above calculation give the static friction force Fr=(1/3)m*g*sinθ.
And maximum firction force is proportion to normal force Fr_max= μ N
Since N=m*g*cosθ.
So μ*m*g*cosθ ≧ (1/3)m*g*sinθ
It means that μ≧(1/3)tanθ
If the angle is increased or μ is not large enough, then the condition for rolling without slipping is not satisified.
It will loss energy due to sliding friction.

So it will be easier to satisfy the condition if static friction coefficient between surface and the rolling object is larger (instead of smaller).

P.S. static friction coefficient is a property between two objects.

[lookang]

what i was trying to highlight the typo error. LOL

the applet is fine, the text in your first post seems wrong

solve the above equation will give you
f=(1/3)m*g*sinθ for cylinder or f=(2/7)m*g*sinθ for sphere.

[Fu-Kwun Hwang]

I have updated the applet as soon as I found the error this morning
(the net force was shown instead of friction force).
 May be your browser still viewing the cached jar file.
Clear your cache , close your broswer and reload it again.
You should be able to view the updated simulation.

I was not reading your message carefully. I thought you were talking about previous error in the simulation.
Thank you. It is fixed now!