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Wize University Physics Textbook (Master) > Angular / Rotational Motion

Rolling Motion

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Rolling without Slipping


Rolling without slipping is a combination of two motions:

1- A transitional motion of the center of the mass with the velocity of:

vcm=R⋅ω\boxed{v_{cm}=R\cdot\omega}

2- A rotational motion of all the points on the object



Watch Out!
In rolling without slipping, the point of contact (point in contact to the surface) is instantaneously at rest! So all points of the rolling object are rotating about this point of contact.

Figure below shows the velocity distribution with respect to distance from the point of contact.







Wize Tip
The rolling without slipping motion is usually caused by the static friction between the point of contact and surface.

PAGE BREAK

Exam Tip
For rolling without slipping:
  • vcm=Rωv_{cm}=R\omega
  • acm=Rαa_{cm}=R\alpha
  • dcm=Rθd_{cm} = R\theta


Energy of a Rolling Object

Since a rolling object has both translational and rotational motion, the total kinetic energy for an object rolling without slipping is

K=12mv2+12Icmω2\boxed{K=\frac12mv^2+\frac12I_{cm}\omega^2}
where the first term is the translational kinetic energy of the center of mass and the second term is the rotational kinetic energy of the object.






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Example: Static Friction for Rolling Down a Slope


Show that the friction for a round object of radius R rolling (without slipping) down an incline of angle θ\thetahas magnitude given by


f=mgsin⁡θ1+mR2If=\dfrac{mg\sin\theta}{1+\dfrac{mR^2}{I}}

Let clockwise rotation be positive, and let the x and y directions be along the slope and perpendicular to the slope. Positive is up for y, and downward along the slope for x.

Solution:

Consider the forces in the x direction:
ΣFx=−fs+mgsin⁡θ=macm\Sigma F_x=-f_s+mg\sin\theta=ma_{cm}
Consider the net torque on the system relative to the center of mass: (Note that only friction has a non-zero torque about the object's center of mass)
Στ=Rfs=Icmαα=RfsIcm\Sigma \tau=Rf_s=I_{cm}\alpha \newline \alpha=\frac{Rf_s}{I_{cm}}
Because the object is rolling without slipping,
acm=Rαa_{cm}=R\alpha
Plugging this into the force equation:
−fs+mgsin⁡θ=m(Rα)−fs+mgsin⁡θ=m(R(RfsIcm))mgsin⁡θ=fs+m(R(RfsIcm))mgsin⁡θ=fs(1+m(R2Icm))fs=mgsin⁡θ1+m(R2Icm)-f_s+mg\sin\theta=m(R\alpha) \newline -f_s+mg\sin\theta=m\Big(R(\frac{Rf_s}{I_{cm}})\Big) \newline mg\sin\theta=f_s+m\Big(R(\frac{Rf_s}{I_{cm}})\Big) \newline mg\sin\theta=f_s(1+m(\frac{R^2}{I_{cm}})) \newline f_s = \dfrac{mg\sin\theta}{1+m(\dfrac{R^2}{I_{cm}})}
Note that in order to have rolling without slipping this static friction should be smaller than the maximum static friction which is equal to μsN\mu_sN. Otherwise, the object will slip during rolling.

Practice: Rolling Marble on a Loop-the-Loop Track


A marble starts from rest and rolls without slipping on the loop-the-loop track in the figure. Find the minimum starting height h from which the ball will remain on the track throughout the loop. Assume the marble's radius is small compared with R. The moment of inertia for a solid sphere is 25Mr2\frac25Mr^2.

Enter your answer by using R = 70 cm (radius of the loop) and r = 1.0 cm (radius of the marble). Your answer should be in meters with three significant figures.
Extra Practice
A spherical ball with a mass of 1kg and radius of 10cm is rolling without slipping down the track shown in the picture starting at rest. (R=2m)
b) What should be the speed of the ball when it gets to the top of the loop to be right at the verge of loosing contact with the track?
A yo-yo is rolling without slipping along the table with a center of mass speed of 10 m/s. A thin string wrapped around the inner hub is attached to a balloon which streams behind the yo-yo. The mass of the balloon is negligible. If the string is coming off the lower side of the inner hub, and the radius of the inner hub is 0.5m and the radius of the outer hub is 0.8m, then the velocity of the balloon with respect to the table is:
→ Vcm=10m/s\rightarrow\ V_{cm}=10m/s
A small solid marble of mass m and radius r rolls without slipping along a loop-the-loop track. From what minimum height above the bottom of the track must the marble be released in order not to leave the track at the top of the loop.
You have two spheres of same mass M and radius R, one solid and one hollow, and an incline on which they can roll without slipping. How can you identify both of them on the incline?
A marble starts from rest and rolls without slipping on the loop-the-loop track in the figure. Find the minimum starting height h from which the ball will remain on the track through the loop. Assume the marble's radius is small compared with R.
Rotational dynamics
A solid, uniform sphere of mass 1.0 kg and radius 1.5 m rolls without slipping down an inclined plane of height 5.0 m. What is the angular velocity of the sphere at the bottom of the inclined plane?
A coin and a ring, each having the same mass and radius are placed at the top of an incline. If both objects are released at the same time, which object will reach the bottom of the incline first?
Enter either coin, ring, both or can't tell as your answer.
Why? (See solution)