# Describing forces in rolling

Consider a wheel on a frictionless horizontal surface. If we apply a horizontal force (parallel to the surface and above the level of the center of mass), what happens to the wheel? Does it roll or slide forward or rotate only or does any other phenomenon happen? Please guide me. Also draw a free body diagram.

Note: This is a thought experiment. If the question is not satisfying, I am sorry for that and please guide me.

Consider the Free Body Diagram:

where:

• $$F$$ is a driving force
• $$mg$$ the weight of the wheel
• $$F_N$$ a reactive force, called the Normal force
• $$F_f$$ a friction force

We can now establish some force/torque balances.

In the vertical ($$y$$) direction, there no motion because with $$\text{N2L}$$:

$$\Sigma F_y=F_N-mg=0 \Rightarrow F_N=mg$$

The friction force is usually modeled as:

$$F_f=\mu F_N=\mu mg$$

As long as no slipping occurs, $$\mu$$ is the static friction coefficient.

Now, looking at the balance of torques about the CoG (marked as $$+$$) we have a net torque balance $$\tau$$:

$$\tau=F\lambda-F_f R=F\lambda-\mu mg R$$

As per $$\text{N2L}$$ (applied for rotation) this causes angular acceleration $$\alpha$$, in the clockwise direction:

$$\tau= I_w \alpha \Rightarrow \alpha=\frac{\tau}{I_w}$$

where $$I_w$$ is the inertial moment of the wheel.

Note that $$\alpha=\frac{\text{d}\omega}{\text{d}t}$$.

Without slipping/sliding, we can write $$v=\omega R$$ and also:

$$a=\alpha R$$

Or:

$$a=\frac{F\lambda-\mu mg R}{I_w}R$$

• I asked for frictionless surface, please. Jun 16 '20 at 1:38
• You CANNOT have rolling without friction. Rolling requires torque to keep the rotational motion going, as also the other answers correctly showed. That torque is supplied by the FRICTION FORCE $F_f=\mu mg$.
– Gert
Jun 16 '20 at 13:18
• On a frictionless surface the wheel will always slide. For that case, simply set $\mu =0$ in the above equations.
– Gert
Jun 16 '20 at 13:24
• I have a question! How do we set the direction of friction? Jun 17 '20 at 2:23
• It's good question and easily answered. A friction force ALWAYS opposes potential or actual movement, including relative movement. So work out which direction movement would happen in the absence of friction and that the opposite direction is the direction the friction force vector will point.
– Gert
Jun 17 '20 at 4:28

It should roll. Whether it will be rolling with sliding or pure rolling we don't know until we know what height the force was applied at. Breaking the motion into Translation and Rotation, we can write one force and torque equation each for both respectively as such-

$$F=ma$$

$$\tau=I\alpha=rF$$

We can also calculate the condition on the height at which you must apply the force for pure rolling by equalling the net acceleration of the bottom most contact point to $$0$$. This point will have two accelerations, one from rotation ($$=R\alpha$$) and one from translation ($$=a$$). Notice their directions are opposite, so for net acceleration of zero they must be equal. So, we have

$$F=ma$$

$$a=\frac{F}{m}$$

$$I\alpha=rF$$

$$\alpha=\frac{rF}{I}$$

For pure rolling, $$a=R\alpha$$

Substituting and rearranging, we have

$$r=\frac{I}{mR}$$

Look neglecting friction makes it simple. We have to think only about the applied force for rotational motion as well as translational motion.

So since there is an external force , the body will have some translational motion in the forward direction. This will happen for sure and we can find the acceleration using

F=ma

Now we have to look for the rotational motion. If the force was applied exactly at the centre of mass of the body, there would be no rotation at all (cause is in your previous question).

But since the force is applied at some distance above the centre of mass there would be some Torque(same cause , as given in your question) and hence the body will rotate.

So in your case , there would be both rotation as well as translation or you can say that the body is rolling. For pure rolling I1mbo has given correct explanation.

Thanks for asking. Hope it helps.

Sorry for no fbd's.

• Please check out this simulation and set coefficient of friction to zero: ophysics.com/r1.html Jun 18 '20 at 2:10
• The body doesn't rotate. And it shouldn't as there will be no Torque . You will need to apply a force ( which you included in your question.)
– user262060
Jun 18 '20 at 7:05

The equation of motions are:

$$\sum_M F_y=m\frac{d}{dt}\,v-F=0$$

$$\sum_M \tau_x=I\,\frac{d}{dt}\,\omega-F\,d=0$$

Thus :

$$\dot{v}=\frac{F}{m}\tag 1$$ $$\dot{\omega}=-\frac{F\,d}{I}\tag 2$$

Roll condition:

$$v=\omega\,R$$

Slide condition :

$$v \lessgtr \omega\,R$$

$$\Rightarrow$$ $$\dot{v}\lessgtr \dot{\omega}\,R\tag 3$$

with equations (1) ,(2) and (3) you get

$$F\left(\frac{1}{m}+\frac{R\,d}{I}\right)\lesseqgtr 0\tag 4$$

where $$=$$ is for roll and > for silde

Thus:

$$F=0$$ for rolling

$$\Rightarrow$$

$$v=v_0\,t\quad,\omega=\frac{v_0}{R}\,t$$ where $$v_0$$ is the initial velocity .

and $$F> 0$$ for slide

• What if v is less than omega r? Jun 29 '20 at 0:54
• Gut question, if the velocity at the contact point is not zero ,you have slid condition I will correct this document, thank you
– Eli
Jun 29 '20 at 7:08