题目

Problem

One end of a light inextensible string is attached to a particle $A$ of mass $5m$ . The other end of the string is attached to a particle $B$ of mass $3m$ . The string passes over a small, smooth, light fixed pulley. Particle $A$ is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4.

Particle $A$ is released.

(a) Find, in terms of $m$ and $g$ , the magnitude of the force exerted on the pulley by the string while $A$ is falling and before $B$ hits the pulley.

(8)

(b) State how, in your solution to part (a), you have used the fact that the pulley is smooth.

(1)

题目中文翻译

一根轻且不可伸长的细绳，一端连接质量为 $5m$ 的质点 $A$ ，另一端连接质量为 $3m$ 的质点 $B$ 。细绳绕过一个光滑轻质小定滑轮。如图 4 所示，质点 $A$ 被保持静止，细绳绷紧，悬挂部分均竖直。

释放质点 $A$ 。

(a) 在 $A$ 下落且 $B$ 尚未碰到滑轮的过程中，用 $m$ 和 $g$ 表示细绳对滑轮施加的力的大小。

(8)

(b) 说明在 (a) 的解答中如何使用了“滑轮光滑”这一条件。

(1)

解答

(a)

解法一

思路

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分别对两个质点沿各自运动方向列方程，先求共同加速度及张力。滑轮两侧的绳都向下拉滑轮，所以合力是两个张力的直接相加。

答题过程

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Let the acceleration magnitude be $a$ and the common tension be $T$ . For $A$ , taking downwards as positive,

5mg-T=5ma.\tag{1}

For $B$ , taking upwards as positive,

T-3mg=3ma.\tag{2}

Adding these equations gives

2mg=8ma,\qquad a=\frac g4.

Substitute into (2):

T=3mg+3m\frac g4=\frac{15mg}{4}.

Both vertical parts of the string pull the pulley downwards, so the force on the pulley has magnitude

F=2T=\boxed{\frac{15mg}{2}}.

(b)

解法一

思路

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回答要对应到滑轮两边的张力相等，并明确提到跨过滑轮的两段绳。

答题过程

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The tension was taken to be the same on both sides of the smooth pulley.

M1 2020 Oct Q7

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程