Rolling (College Board AP® Physics 1: Algebra-Based): Study Guide

Total kinetic energy

• The total kinetic energy of a system is the sum of translational and rotational kinetic energies:

total kinetic energy = translational kinetic energy + rotational kinetic energy

$K_{total}=K_{trans}+K_{rot}$

$K_{total}=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

• Where:

  • $m$ = mass of the object, in $\mathrm{kg}$

  • $v$ = linear velocity, in $\mathrm{m}/\mathrm{s}$

  • $I$ = rotational inertia, in $\mathrm{kg}·{\mathrm{m}}^2$

  • $\omega$ = angular velocity, in $\mathrm{rad}/\mathrm{s}$

• When considering the mechanical energy of a system, the kinetic energy can be broken down into its translational and rotational components

$E=∆K+\hspace{0.17em}∆U$

$E=∆\left(K_{trans}+K_{rot}\right)+\hspace{0.17em}∆U$

• Two special cases involving rotational kinetic energy are

  • rolling motion without slipping

  • rolling motion while slipping

Rolling without slipping

• Rolling motion is a combination of rotating and sliding (translational) motion

• When a circular rigid body (e.g. a ball, wheel, or cylinder) of radius $r$ rolls without slipping:

  • the translational motion of the system's center of mass is directly related to its rotational motion

  • the point of contact between the rotating body and the surface is at rest

• The distance traveled by the system's center of mass is:

$∆x_{cm}=r∆\theta$

• Where:

  • $∆x_{cm}$ = translational distance moved by the object's center of mass, in $\mathrm{m}$

  • $∆\theta$ = angular displacement of the rolling object, in $\mathrm{rad}$

• The linear velocity of the system's center of mass is:

$v_{cm}=r\omega$

• Where:

  • $v_{cm}$ = translational velocity of the object's center of mass, in $\mathrm{m}/\mathrm{s}$

  • $\omega$ = angular speed of the rolling object, in $\mathrm{rad}/\mathrm{s}$

• The linear acceleration of the system's center of mass is:

$a_{cm}=r\alpha$

• Where:

  • $a_{cm}$ = translational acceleration of the object's center of mass, in $\mathrm{m}/{\mathrm{s}}^2$

  • $\alpha$ = angular acceleration of the rolling object, in $\mathrm{rad}/{\mathrm{s}}^2$

• These equations appear to be identical to the equations for fixed-axis rotation, but there are some key differences

  • For fixed axis rotation: $s$, $v$, and $a$ refer to the tangential quantities at a point on the object

  • For a rolling object: $∆x_{cm}$, $v_{cm}$, and $a_{cm}$ refer to the translational quantities of the object's center of mass

Translational and rotational components of a rotating wheel

The role of friction in rolling without slipping

• When an object rolls without slipping, static friction causes the rotation about the center of mass

• This happens when there is no relative motion between the surface and the point of contact

• The direction of the frictional force always acts to oppose the potential slipping motion

Static friction causing rolling on a horizontal plane

Static friction causing rolling on an inclined plane

• In the case of rolling motion without slipping:

  • no kinetic friction is present

  • only static friction is present

• Since no work is done by static friction, no energy is dissipated when a system rolls without slipping

Rolling on a horizontal surface without slipping

• When a disk rotates on a horizontal surface:

  • all points on the disk have a constant angular speed

  • each point on the disc has a different linear velocity depending on its distance from the center $\left(v\propto r\right)$

  • the linear speed is the same at all points on the circumference

• When a disk slips or slides on a horizontal surface:

  • there is not enough static friction present to allow the object to roll

  • all points on the disk have the same linear velocity

  • the angular velocity is zero

• So, when a disk rolls without slipping:

  • there is enough static friction present to initiate rotational motion allowing the object to roll

  • the point in contact with the surface has a velocity of $v=0$

  • the center of mass has a velocity of $v_{cm}=\omega r$

  • the top point has a velocity of $v=2\omega r$, or $2v$

Rolling on an inclined plane without slipping

• Another common scenario is a circular rigid body rolling on an inclined plane without slipping

• Since the object rolls without slipping, no work is done by frictional forces

• Therefore, the total mechanical energy of the system is conserved

  • At the top of the slope, the object has gravitational potential energy only (assuming it is initially at rest)

  • As the object rolls down the slope, the gravitational potential energy is transferred to kinetic energy

  • At the bottom of the slope, the object has both translational and rotational kinetic energy

• The linear or angular velocity can then be determined by:

  • applying energy conservation laws

  • using the equation for the rotational inertia of the object

  • using the relationship between linear and angular velocity $v_{cm}=\omega r$

Rolling while slipping

• When a rigid body rolls while slipping, the translational motion of the system's center of mass cannot be directly related to its rotational motion

$∆x_{cm}\ne r∆\theta$

$v_{cm}\ne r\omega$

$a_{cm}\ne r\alpha$

• This is because the point of contact between the rotating body and the surface is not at rest

• In the case of rolling motion while slipping:

  • there is not enough static friction to cause the rotation about the system's center of mass

  • only kinetic friction is present as there is relative motion between the surface and the point of contact

• As with the no-slipping case, the direction of the frictional force acts to oppose the slipping motion

• Since work is done by kinetic friction, energy is dissipated from the system when slipping occurs

• The amount of energy dissipation (work done) depends on

  • the coefficient of kinetic friction

  • the distance over which slipping occurs