架飞机以与水平面成.pptVIP

The Pre-Olympiad Workshop Tutorial Mechanics * Example 1: An airplane is initially rising up at speed v0 at an angle ? to the horizon. Find the trajectory of the plane such that weightless condition can be achieved in the plane. 一架飛機以與水平面成? 角的初速度v0上升。求飛 機以什麼樣的軌跡飛行，能使飛機裏的物體處於失 重狀態。 q v0 Solution: People inside the airplane will observe weightlessness when it is under free fall. ? Constant downward acceleration g In horizontal direction: In vertical direction (upward = positive): Apply The plane should follow the parabola. 飛機須沿抛物線運動。 Example 2: As shown, two identical weights are fixed on the two ends of a uniform rigid rod of length L. The upper weight is restricted to move on a smooth horizontal rail and the rod is free to swing along the rail. The masses of the weights and the rod are equal. Find the small angle vibration frequency of the system. 如圖所示，兩個質量為m的重塊分別固定在一根長度 為L質量為m的均勻杆兩端。上面的重塊可以沿光滑的 水平軌道滑行，杆可沿軌道方向自由擺動。求整個系 統的小角度振動頻率。 . Solution: No horizontal external force ? The center of the rod will not move in the horizontal direction杆中心在水平方向不動。 Use The forces acting upon the rod are shown. The torque to the center of the rod is 由如圖力的分析，可得 . . Finally, 最後得 Example 3: A uniform iron pulley of total mass M1 and radius R is fixed above the ground. It can spin freely about its center. Additional weights in the form of many small magnets with total mass M2 can be attached to the pulley. A long thread is winding around its edge, and at its end hangs a weight of mass M3 at height H from the ground that can be released suddenly. Assume that there is no slip between the thread and the pulley. (a) How should the magnets be placed with circular symmetry on the pulley such that the final speed of M3 is minimum when it reaches the ground? (b) Find the minimum speed. 一均勻鐵滑輪質量為M1，半徑為R，固定在地面上方。滑輪可以輪心自由轉動。 另有總重量為M2的許多小磁鐵塊可吸附在滑輪上。一長細繩繞著滑輪邊緣，終 端掛一質量為M3的小重塊，離地高度為H。細繩與滑輪間無滑動。 (a)問小磁鐵塊應怎樣圓對稱地分佈在滑輪上，才能使小重塊被放開後到達地面 時的速度為最小？ (b) 求該速度。 Solution: When the moment of inerti