11.4
滑らずに転がる
車輪が発明されて以来、人々は滑ることなく転がる動きを観察してきました。 たとえば、車のタイヤと路面の相互作用を観察できます。 ドライバーがアクセルを床に押し付けて、車が前に進まないでタイヤが回転する場合、車輪と路面の間に動摩擦が発生するはずです。 ドライバーがゆっくりアクセルを踏んで車を前進させると…
物体が摩擦のない表面で加速せずに純粋な並進運動を行うと、2つの力だけが作用します。
物体を回転させるには、トルクが必要です。摩擦が導入されると、摩擦力は線速度の方向と反対に作用し、トルクが生成されます。
このトルクによって角加速度が発生し、物体が回転し始めます。回転速度が十分に大きくなると、物体の角速度は、表面と接触する点の接線速度を相殺するのに十分になります。
この点の接線速度は等しく、重心の接線方向と反対方向であるため、速度はゼロになります。この動きを滑らずに転がるという
。ここでは、オブジェクトの重心は線形パスをたどりますが、オブジェクトの縁上のポイントはサイクロイド パスをたどります。
オブジェクトが 1 回完全に回転すると、重心は転がるオブジェクトの円周に等しい直線距離だけ移動します。
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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.