6.4
第二定律：相同加速度下的运动
牛顿第二运动定律适用于在相同加速度下运动的物体。 例如，当行李牵引车牵引行李车时，每辆行李车都以与牵引车相同的加速度移动。
考虑在无摩擦的水平表面上具有一定质量的块 1，该块被穿过无摩擦且无质量的滑轮的轻绳拉动。 绳子的另一端连接到表面边缘上方不同质量的块 2。 在这里，应用牛顿第二定律来计算弦中的…
以相同加速度运动的物体遵循牛顿第二定律。考虑一辆汽车通过一根无质量且不可伸长的绳索拖拽另一辆完全相同的汽车。绳索对两辆汽车施加大小相等、方向相反的力。那么，这辆汽车为何会产生加速度？
为回答此问题，需绘制每个物体的受力分析图。由于竖直方向的力相互平衡，因此每个物体所受的合力可通过水平方向的力分量求得。
现在，利用两辆汽车质量相同且加速度相同的事实，将方程联立，得到使汽车加速的力。
考虑另一个例子：一根无质量且不可伸长的绳子跨过一个无摩擦的滑轮，悬挂着两个不相等的质量。当系统释放后，较重的物体将较轻的物体提起，两者以相同的加速度运动。
通过受力分析图和牛顿第二定律，可以推导出每个物体所受合外力的方程。将这两个方程相减并重新整理后，即可求得物体的加速度值。
View the full transcript and gain access to JoVE Core videos
Q1: Why do two cars connected by a rope accelerate together?
When a car tows another identical car with a massless inextensible rope, both experience the same acceleration because the rope applies equal and opposite forces on each. Using free-body diagrams, the net horizontal force on each car equals its mass times acceleration. Since both cars have identical mass and acceleration, they move together as a system.
Q2: How do you find acceleration when unequal masses hang over a pulley?
Draw free-body diagrams for each mass and apply Newton's second law separately. The heavier mass pulls the lighter one upward while descending. Both masses move with the same acceleration despite their different weights. Subtract the force equations and rearrange to solve for the acceleration value.
Q3: What role does tension play in a string connecting two blocks?
Tension is the force transmitted through a massless string connecting two blocks. The tension remains constant throughout the string and acts horizontally on a block on a surface and vertically on a hanging block. Tension pulls each block in opposite directions, enabling them to move with the same acceleration.
Q4: Why do blocks on a horizontal surface and hanging blocks have equal acceleration?
When one block moves horizontally on a frictionless surface and another hangs vertically, connected by a string over a pulley, they travel equal distances in equal times. This constraint means both blocks must have the same acceleration magnitude. Newton's second law applied to each block confirms this kinematic relationship.
Q5: How do free-body diagrams help solve problems with same acceleration?
Free-body diagrams isolate each object and show all forces acting on it. For objects with the same acceleration, you write component equations for horizontal and vertical forces separately. Balanced forces in one direction simplify the analysis, allowing you to focus on net forces causing acceleration.
Q6: What happens to tension when two blocks of different masses hang vertically?
When unequal masses hang vertically over a frictionless pulley, the tension in the string is less than the weight of the heavier mass but greater than the weight of the lighter mass. The tension accelerates the lighter mass upward and reduces the net downward force on the heavier mass, allowing both to accelerate together.
Q7: Why are vertical and horizontal forces analyzed separately in these problems?
Forces perpendicular to each other do not affect one another. On a horizontal surface, vertical forces balance, so only horizontal components determine acceleration. For hanging blocks, only vertical forces matter. Separating components simplifies calculations and reveals which forces actually cause the observed acceleration.