使用中文,动画演示要作为重点。运算过程要配合图形解释---**Question 8 (3 points)**
**Question Stem:**
工人在河堤的硬质坡面上固定一垂直坡面的挡板,向坡底运送长方体建筑材料。如图所示,坡面与水平面夹角为θ,交线为PN,坡面内QN与PN垂直,挡板平面与坡面的交线为MN,∠MNQ=θ。若建筑材料与坡面、挡板间的动摩擦因数均为μ,重力加速度大小为g,则建筑材料沿MN向下滑动的加速度大小为 ( )
**Translation of Question Stem:**
Workers fix a baffle (挡板) perpendicular to the hard slope surface of a river embankment to transport rectangular building materials to the bottom of the slope. As shown in the figure, the angle between the slope surface (坡面) and the horizontal plane is θ, their intersection line is PN. QN in the slope plane is perpendicular to PN. The intersection line between the baffle plane and the slope plane is MN, and ∠MNQ=θ. If the kinetic friction coefficient between the building material and the slope surface, and between the building material and the baffle, are both μ, and the acceleration due to gravity is g, what is the magnitude of the acceleration of the building material sliding down along MN?
**Options:**
A. $g\sin^2\theta - \mu g\cos\theta - \mu g\sin\theta\cos\theta$
B. $g\sin\theta\cos\theta - \mu g\cos\theta - \mu g\sin^2\theta$
C. $g\sin\theta\cos\theta - \mu g\cos\theta - \mu g\sin\theta\cos\theta$
D. $g\cos^2\theta - \mu g\cos\theta - \mu g\sin^2\theta$
**Chart/Diagram Description:**
* **Type:** A 3D perspective view illustrating a setup with two intersecting planes and a rectangular block.
* **Main Elements:**
* **Planes:**
* A main inclined plane (坡面) is depicted. Its base line, where it intersects the horizontal plane (implied), is labeled PN. The angle of inclination of this slope with respect to the horizontal is denoted by θ.
* A second plane, described as a "baffle" (挡板), is shown intersecting the main inclined plane. This baffle plane is stated to be perpendicular to the slope surface itself.
* **Lines and Points:**
* **P, N, M, Q:** Labeled points define the geometry.
* **PN:** Represents the intersection line of the inclined plane with the horizontal base.
* **QN:** A line on the inclined plane. It is stated that QN is perpendicular to PN. This implies that QN is a line of steepest descent on the inclined plane. The angle between QN and the horizontal is θ (the slope angle).
* **MN:** Represents the intersection line between the inclined plane and the baffle plane. The rectangular block slides along this line.
* **MQ:** A line on the baffle plane connecting M and Q.
* **Angles:**
* An angle labeled θ is shown at N, representing the angle between the inclined plane (specifically, the line QN) and the horizontal. This is the slope angle.
* Another angle labeled θ is shown at N, specifically ∠MNQ. This angle is formed by the line MN (the sliding path) and the line QN (the steepest descent line on the slope) within the plane of the slope.
* **Object:** A small rectangular block (建筑材料) is placed in the corner formed by the intersection of the two planes (the main inclined plane and the baffle), resting on both surfaces.
* **Shading:** The triangular region QNM, representing the baffle, is shaded. Another triangular region with base QN and a vertical side is also shaded, emphasizing the vertical height associated with the slope.
* **Relative Position:** The block is shown near the top part of the intersection line MN, implying it will slide downwards along MN.
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这是一个关于斜坡挡板的物理问题。建筑材料放置在斜坡面和垂直挡板形成的夹角中。坡面与水平面夹角为θ,挡板与坡面垂直。材料将沿着MN线向下滑动。我们需要分析材料在重力和摩擦力作用下的运动。
现在分析建筑材料的受力情况。材料受到重力mg的作用,这个重力需要分解到三个方向。沿着MN滑动方向的分力为mg乘以sin theta乘以cos theta。垂直于坡面向下的分力为mg cos theta。垂直于挡板的分力为mg sin平方theta。这些分力将决定材料的运动状态。
接下来分析摩擦力。材料与坡面接触产生正压力N1等于mg cos theta,对应的摩擦力f1等于μ乘以mg cos theta。材料与挡板接触产生正压力N2等于mg sin平方theta,对应的摩擦力f2等于μ乘以mg sin平方theta。这两个摩擦力都与材料的运动方向相反,阻碍材料沿MN方向滑动。
现在应用牛顿第二定律来求解加速度。沿MN方向的驱动力为mg sin theta cos theta。阻碍运动的摩擦力有两个:坡面摩擦力μmg cos theta和挡板摩擦力μmg sin平方theta。根据牛顿第二定律,合力等于质量乘以加速度,因此加速度等于g sin theta cos theta减去μg cos theta再减去μg sin平方theta。这就是选项B的表达式。
让我们确认答案。通过完整的受力分析和牛顿第二定律的应用,我们得到了建筑材料沿MN方向滑动的加速度表达式:a等于g sin theta cos theta减去μg cos theta再减去μg sin平方theta。对比四个选项,这个表达式与选项B完全一致。因此正确答案是B。