题目

Problem

A hollow toy is formed by joining a uniform right circular conical shell $C$ , with radius $4a$ and height $3a$ , to a uniform hemispherical shell $H$ , with radius $4a$ . The circular edge of $C$ coincides with the circular edge of $H$ , as shown in Figure 1.

The mass per unit area of $C$ is $\lambda$ and the mass per unit area of $H$ is $k\lambda$ where $k$ is a constant.

Given that the centre of mass of the toy is a distance $4a$ from the vertex of the cone, find the value of $k$ .

(6)

题目中文翻译

将半径为 $4a$ 、高为 $3a$ 的均匀正圆锥壳 $C$ 与半径为 $4a$ 的均匀半球壳 $H$ 连接，组成一个空心玩具。如图 1 所示， $C$ 的圆形边缘与 $H$ 的圆形边缘重合。

$C$ 的单位面积质量为 $\lambda$ ， $H$ 的单位面积质量为 $k\lambda$ ，其中 $k$ 为常数。

已知玩具质心到圆锥顶点的距离为 $4a$ ，求 $k$ 的值。

(6)

解答

解法一

思路

展开

两个部分都是薄壳，质量应由曲面面积乘面密度求得。分别找出圆锥壳、半球壳的轴向质心位置，再统一从圆锥顶点取质量矩。

答题过程

展开

The slant height of the cone is

s=\sqrt{(4a)^2+(3a)^2}=5a.

The conical shell has curved area $\pi rs=20\pi a^2$ and mass

M_C=20\pi a^2\lambda.

The hemispherical shell has curved area $2\pi r^2=32\pi a^2$ and mass

M_H=32\pi a^2k\lambda.

For a uniform conical shell, the centre of mass is $2h/3$ from its vertex, so it is $2a$ from the vertex here. A uniform hemispherical shell has its centre of mass $r/2=2a$ from the centre of its plane face, into the hemisphere. Thus its distance from the cone’s vertex is $3a+2a=5a$ . Taking mass moments about the vertex along the common axis,

20\pi a^2\lambda(2a)+32\pi a^2k\lambda(5a)=(20+32k)\pi a^2\lambda(4a).

Cancelling $4\pi a^3\lambda$ ,

10+40k=4(5+8k),\qquad 8k=10.

Therefore

\boxed{k=\frac54}.

M3 2021 June Q1

题目

解答

解法一

思路

答题过程