2010年12月23日 星期四

純滾動

純滾動與圓周運動
(有動畫顯示)

半徑為 R的圓形物體以角速率ω轉動時,其圓周相對於圓心的瞬時速率為 v = ωR 若圓形的物體沿著平面運動,且物體質心相對該平面的相對速率也是 v時, 則該物體與平面接觸點的相對速度恆為零時,稱該物體為純滾動.
追蹤該物體圓周上固定一點形成的軌跡,稱為擺線.

wiki:

純滾動

假若,一個圓柱形物體滾動(rolling)於平面上,不做任何滑動運動(物體與平面之間,沒有任何滑動摩擦),則這物體的運動稱為純滾動,其質心的速度  \boldsymbol{ v}_G\, 必須符合約束條件:
\boldsymbol{ v}_G(t) = \boldsymbol{\omega} \times \boldsymbol{R}_{G/O}\, ;
其中,\boldsymbol{\omega}\, 是旋轉的角速度,\boldsymbol{R}_{G/O}\, 是從物體接觸平面的接觸點到物體質心的位移向量。
對於物體在滾動時不傾斜或不轉彎的案例,這約束條件約化為
v = R \omega\, 。



補充:擺線軌跡的單擺

2010年12月3日 星期五

Physics Chapter8-64

P.230

64. A block of mass m1 = 20.0 kg is connected to a block of mass m2 = 30.0 kg by a massless string that passes over a light, frictionless pulley. The 30.0-kg block is connected to a spring that has negligible mass and a force constant of k = 250 N/m as shown in Figure P8.64. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0-kg block is pulled a distance h = 20.0 cm down the incline of angle q = 40.0° (so that the 30.0-kg block is 40.0 cm above the floor) and released from rest. Find the speed of each block when the 30.0-kg block is 20.0 cm above the floor (that is, when the spring is unstretched).