题目

Problem

Two particles, $P$ and $Q$ , have masses $4m$ and $2m$ respectively. The particles are connected by a light inextensible string. A second light inextensible string has one end attached to $Q$ . Both strings are taut and vertical, as shown in Figure 1.

The particles are accelerating vertically downwards.

Given that the tension in the string connecting the two particles is $3mg$ , find, in terms of $m$ and $g$ , the tension in the upper string.

(6)

题目中文翻译

两质点 $P$ 和 $Q$ 的质量分别为 $4m$ 和 $2m$ ，由一根不可伸长的轻绳相连。另一根不可伸长的轻绳的一端连接于 $Q$ 。如图 1 所示，两根绳均绷紧且竖直。

两质点正在竖直向下加速运动。

已知连接两质点的绳中张力为 $3mg$ ，用 $m$ 和 $g$ 表示上方绳中的张力。

(6)

解答

解法一

思路

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对下方的 $P$ 先求共同加速度，再对上方 $Q$ 列方程。下方绳对 $Q$ 的拉力向下，不能漏掉或误作向上。

答题过程

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Let $a$ be the common downward acceleration and $T$ the tension in the upper string. The taut inextensible connecting string gives both particles the same acceleration.

For the lower particle $P$ , taking downwards as positive,

4mg-3mg=4ma,\qquad a=\frac g4.

For the upper particle $Q$ , its weight and the lower string tension both act downwards, while $T$ acts upwards. Thus

\begin{align*} 2mg+3mg-T=&\,2ma,\\ 5mg-T=&\,2m\frac g4,\\ T=&\,\boxed{\frac{9mg}{2}}. \end{align*}

M1 2024 June Q3

题目

解答

解法一

思路

答题过程