11.4
Rolling Without Slipping
People have observed the rolling motion without slipping ever since the invention of the wheel. For example, one can look at the interaction between a…
When an object performs pure translational motion without acceleration on a frictionless surface, only two forces act on it.
For the object to spin, torque is required. If friction is introduced, the frictional force acts opposite to the direction of the linear velocity, producing the torque.
This torque produces an angular acceleration, and the object starts spinning. As the spinning speed becomes sufficiently large, the angular speed of the object is enough to cancel the tangential speed of the point in contact with the surface.
The tangential velocity of this point is equal and in the opposite direction to that of the center of mass, resulting in zero velocity. This motion is called rolling without slipping.
Here, the center of mass of the object follows a linear path, but the point on the rim of the object follows a cycloid path.
In one complete rotation of the object, the center of mass moves by the linear distance equal to the circumference of the rolling object.
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Q1: What role does friction play in rolling without slipping?
Friction is essential for rolling without slipping. When an object moves forward with spinning, static friction between the object and surface produces net torque, causing angular acceleration. As the object spins faster, the tangential velocity of the contact point eventually equals and opposes the center of mass velocity, resulting in zero relative velocity at the contact point.
Q2: How does the center of mass move during rolling without slipping?
During rolling without slipping, the center of mass follows a linear path while the contact point on the rim traces a cycloid path. In one complete rotation, the center of mass advances a distance equal to the object's circumference. This relationship connects linear and angular motion through the rolling condition.
Q3: What is the difference between rolling with slipping and rolling without slipping?
In rolling with slipping, kinetic friction acts between the object and surface, but the contact point moves relative to the ground. In rolling without slipping, static friction acts and the contact point remains at rest relative to the ground. The key distinction is whether relative motion exists at the contact surface.
Q4: What condition must be met for an object to roll without slipping?
Rolling without slipping occurs when the tangential velocity of the contact point equals and opposes the linear velocity of the center of mass, producing zero relative velocity. Mathematically, the center of mass velocity equals the radius multiplied by angular velocity. This condition requires sufficient angular acceleration from friction-generated torque.
Q5: Why is static friction present when a tire rolls without slipping?
Static friction is present because the contact point between the tire and road surface is momentarily at rest relative to the ground during rolling without slipping. Even though the tire rotates and moves forward, the bottom surface does not slip. This static friction generates the net torque needed to maintain the spinning motion.
Q6: How does torque develop during the transition to rolling without slipping?
Initially, friction acts opposite to the object's linear velocity, producing torque that causes angular acceleration. As the object spins faster, angular velocity increases until the tangential velocity of the rim point matches the center of mass velocity. This torque generation is governed by the equation of rotational dynamics, which relates net torque to angular acceleration.
Q7: What path does a point on the rim trace during rolling without slipping?
A point on the rim traces a cycloid path, a curve generated by a point on a circle rolling along a straight line. While the center of mass moves linearly, each rim point follows this distinctive looped trajectory. The cycloid path demonstrates the combined effect of the object's rotation and translation during rolling without slipping.