11.4
Rolagem sem deslizar
As pessoas têm observado o movimento de rolagem sem deslizar desde a invenção da roda. Por exemplo, podemos observar a interação entre os pneus de um…
Quando um objeto executa movimento de translação puro sem aceleração em uma superfície sem atrito, apenas duas forças atuam sobre ele.
Para que o objeto gire, é necessário torque. Se o atrito for introduzido, a força de atrito atua na direção oposta à direção da velocidade linear, produzindo o torque.
Esse torque produz uma aceleração angular e o objeto começa a girar. À medida que a velocidade de rotação se torna suficientemente grande, a velocidade angular do objeto é suficiente para cancelar a velocidade tangencial do ponto em contato com a superfície.
A velocidade tangencial deste ponto é igual e na direção oposta à do centro de massa, resultando em velocidade zero. Este movimento é chamado de rolar sem escorregar.
Aqui, o centro de massa do objeto segue um caminho linear, mas o ponto na borda do objeto segue um caminho ciclóide.
Em uma rotação completa do objeto, o centro de massa se move pela distância linear igual à circunferência do objeto rolante.
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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.