题目

Problem

Figure 1 shows a hollow cylinder with diameter $70\,\mathrm{cm}$ . The cylinder is rotating at a constant angular speed about its axis, which is vertical. As the cylinder rotates, a particle $P$ remains in contact with the same point on the rough inside surface of the cylinder. The particle is moving in a horizontal circle of diameter $70\,\mathrm{cm}$ .

The cylinder makes $2$ complete revolutions every second.

The coefficient of friction between $P$ and the cylinder is $\mu$ .

Find the range of possible values of $\mu$ .

(6)

题目中文翻译

图 1 所示的空心圆筒的直径为 $70\,\mathrm{cm}$ 。圆筒绕其竖直轴以恒定角速度转动。圆筒转动时，质点 $P$ 始终与圆筒粗糙内表面上的同一点接触。质点在直径为 $70\,\mathrm{cm}$ 的水平圆周上运动。

圆筒每秒转动 $2$ 整圈。

$P$ 与圆筒之间的摩擦系数为 $\mu$ 。

求 $\mu$ 的可能取值范围。

(6)

解答

解法一

思路

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质点没有竖直加速度，向上的静摩擦力平衡重力；圆筒的支持力水平指向轴心，提供向心力。静摩擦不一定达到最大值，因此最后得到摩擦系数的不等式。

答题过程

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Let the mass of $P$ be $m$ , the normal reaction be $R$ , and the upward frictional force be $F$ . The radius of the circular path is $0.35$ m. The cylinder makes $2$ revolutions per second, so

\omega=2(2\pi)=4\pi\text{ rad s}^{-1}.

Resolving horizontally towards the axis,

R=m\omega^2r=m(4\pi)^2(0.35)=\frac{28}{5}m\pi^2.

There is no vertical acceleration, so

F=mg.

For no vertical slipping, the required friction must not exceed its limiting value:

F\leq\mu R,\qquad mg\leq\mu\frac{28}{5}m\pi^2.

Hence

\mu\geq\frac{5g}{28\pi^2}.

Using $g=9.8$ , the range is

\boxed{\mu\geq0.177}\quad(3\text{ s.f.}).

Equality is possible when the friction is limiting.

M3 2020 Oct Q1

题目

解答

解法一

思路

答题过程