题目

Problem

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $5a$

The other end of the string is attached to a fixed point $A$ above a smooth horizontal table.

The particle moves in a horizontal circle on the table with the string taut.

The centre $O$ of the circle is vertically below $A$ with $OA=4a$ , as shown in Figure 1.

The particle moves with constant angular speed $\omega$ and remains in contact with the table.

(a) Find, in terms of $m$ and $g$ , the magnitude of the normal reaction between $P$ and the table when $\omega=\dfrac14\sqrt{\dfrac ga}$

(6)

The angular speed $\omega$ is gradually increased and the particle remains in contact with the table.

(b) Show that $\omega<\dfrac12\sqrt{\dfrac ga}$

(2)

题目中文翻译

质量为 $m$ 的质点 $P$ 连接于一根长为 $5a$ 的不可伸长轻绳的一端，另一端连接于光滑水平桌面上方的固定点 $A$ 。

质点在桌面上做水平圆周运动，绳保持绷紧。

如图 1 所示，圆心 $O$ 位于 $A$ 正下方，且 $OA=4a$ 。

质点以恒定角速度 $\omega$ 运动，并保持与桌面接触。

(a) 当 $\omega=\dfrac14\sqrt{\dfrac ga}$ 时，用 $m$ 和 $g$ 表示 $P$ 与桌面之间的法向反作用力大小。

(6)

逐渐增大角速度 $\omega$ ，质点仍与桌面接触。

(b) 证明 $\omega<\dfrac12\sqrt{\dfrac ga}$ 。

(2)

解答

(a)

解法一

思路

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先由 $3:4:5$ 三角形确定水平圆周半径和张力分量。桌面支持力与张力的竖直分量共同平衡重力。

答题过程

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The horizontal radius is

OP=\sqrt{(5a)^2-(4a)^2}=3a.

Let $T$ be the tension and let $\beta$ be the angle the string makes with the horizontal. Then $\cos\beta=3/5$ and $\sin\beta=4/5$ .

Resolving horizontally towards $O$ ,

\frac35T=m(3a)\omega^2, \qquad T=5ma\omega^2.

With $\omega=\frac14\sqrt{g/a}$ ,

T=5ma\frac{g}{16a}=\frac{5mg}{16}.

If $R$ is the normal reaction, vertical equilibrium gives

R+\frac45T=mg.

Consequently

\boxed{R=mg-\frac45\frac{5mg}{16}=\frac34mg.}

(b)

解法一

思路

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把角速度保留为变量，支持力会随角速度增大而减小。正支持力对应仍由桌面支承；支持力为零是失去接触的临界情况。

答题过程

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For a general angular speed, part (a) gives $T=5ma\omega^2$ . Thus

R=mg-\frac45T=mg-4ma\omega^2.

Before contact is lost, the normal reaction is positive. Hence

mg-4ma\omega^2>0, \qquad \omega^2<\frac{g}{4a}.

Since $\omega$ is a non-negative angular speed,

\boxed{\omega<\frac12\sqrt{\frac ga}.}

At equality, $R=0$ : this is the limiting condition for losing contact with the table.

M3 2026 Jan Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程