题目

Problem

A metal girder $AB$ has weight $W$ newtons and length $6\,\mathrm{m}$ . The girder rests in a horizontal position on two supports $C$ and $D$ where $AC=DB=1\,\mathrm{m}$ , as shown in Figure 2.

When a force of magnitude $900\,\mathrm{N}$ is applied vertically upwards to the girder at $A$ , the girder is about to tilt about $D$ .

When a force of magnitude $1500\,\mathrm{N}$ is applied vertically upwards to the girder at $B$ , the girder is about to tilt about $C$ .

The girder is modelled as a non-uniform rod whose centre of mass is a distance $x$ metres from $A$ .

Find the value of $x$ .

(6)

题目中文翻译

一根金属梁 $AB$ 重 $W$ 牛顿，长 $6\,\mathrm{m}$ 。如图 2 所示，梁水平放置在支点 $C$ 和 $D$ 上，其中 $AC=DB=1\,\mathrm{m}$ 。

当在 $A$ 点对梁施加大小为 $900\,\mathrm{N}$ 的竖直向上的力时，梁恰好要绕 $D$ 点倾斜。

当在 $B$ 点对梁施加大小为 $1500\,\mathrm{N}$ 的竖直向上的力时，梁恰好要绕 $C$ 点倾斜。

将梁视为非均匀杆，其质心距 $A$ 点 $x$ 米。

求 $x$ 的值。

(6)

解答

解法一

思路

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第一次将绕 $D$ 倾斜， $C$ 处支持力为零；第二次将绕 $C$ 倾斜， $D$ 处支持力为零。分别关于转轴取矩，再消去未知重力。

答题过程

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The supports are $1\,\mathrm{m}$ and $5\,\mathrm{m}$ from $A$ .

With the upward force of $900\,\mathrm{N}$ at $A$ , the reaction at $C$ is zero. Taking moments about $D$ ,

900(5)=W(5-x).\tag{1}

With the upward force of $1500\,\mathrm{N}$ at $B$ , the reaction at $D$ is zero. Taking moments about $C$ ,

1500(5)=W(x-1).\tag{2}

Dividing (2) by (1),

\frac{x-1}{5-x}=\frac{7500}{4500}=\frac53.

Therefore

3x-3=25-5x, \qquad 8x=28, \qquad \boxed{x=3.5}.

The centre of mass is $3.5\,\mathrm{m}$ from $A$ , between the two supports. Substitution gives $W=3000\,\mathrm{N}$ , consistent with both limiting moment equations.

M1 2021 Jan Q4

题目

解答

解法一

思路

答题过程