题目

Problem

One end of a light inextensible string of length $a$ is attached to a fixed point $O$ which lies above a smooth horizontal table.

The other end is attached to a particle $P$ of mass $m$ .

The particle $P$ moves on the table in a horizontal circle with constant angular speed $\sqrt{\dfrac g{2a}}$ and with the string taut.

The string makes a constant angle of $30^\circ$ with the table, as shown in Figure 2.

Find, in terms of $m$ and $g$ , the magnitude of the force exerted on $P$ by the table.

(7)

题目中文翻译

一根长为 $a$ 的不可伸长轻绳的一端连接于光滑水平桌面上方的固定点 $O$ ，另一端连接于质量为 $m$ 的质点 $P$ 。

质点 $P$ 在桌面上以恒定角速度 $\sqrt{\dfrac g{2a}}$ 做水平圆周运动，绳保持绷紧。

如图 2 所示，绳与桌面的夹角恒为 $30^\circ$ 。

用 $m$ 和 $g$ 表示桌面对 $P$ 施加的力的大小。

(7)

解答

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角度从桌面量起，所以水平分量用余弦。先由水平向心方程求张力，再由竖直平衡求桌面支持力，不能直接令支持力等于重力。

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Let $T$ be the tension and $R$ the upward normal force from the table. The horizontal radius is

r=a\cos30^\circ,

and $\omega^2=g/(2a)$ . Resolving horizontally towards the centre of the circle,

T\cos30^\circ=mr\omega^2 =m(a\cos30^\circ)\frac{g}{2a}.

Cancelling $\cos30^\circ$ gives

T=\frac12mg.

There is no vertical acceleration, so

R+T\sin30^\circ=mg.

Thus

\boxed{R=mg-\frac12mg\cdot\frac12=\frac34mg.}