题目

Problem

A suitcase of mass $40\,\mathrm{kg}$ is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap.

The strap is inclined at $20^\circ$ above the horizontal.

The coefficient of friction between the suitcase and the floor is $\frac34$

The strap is modelled as a light inextensible string and the suitcase is modelled as a particle.

Find the tension in the strap.

(7)

题目中文翻译

用一条细带拉着质量为 $40\,\mathrm{kg}$ 的行李箱，使其沿粗糙水平地板以恒定速率作直线运动。

细带朝水平面上方倾斜，与水平面成 $20^\circ$ 角。

行李箱与地板之间的摩擦系数为 $\frac34$ 。

将细带视为不可伸长的轻绳，将行李箱视为质点。

求细带中的张力。

(7)

解答

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行李箱在水平地板上匀速运动，水平与竖直合力均为零。细带向上的分力减小支持力，不能把细带倾角误认为地面倾角。

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Let the tension be $T$ , the normal reaction $R$ , and the friction $F$ .

Resolving vertically,

R+T\sin20^\circ=40g,

R=392-T\sin20^\circ.

Since the suitcase is moving at constant speed, horizontal equilibrium gives

T\cos20^\circ=F=\frac34R.

Substituting for $R$ ,

\begin{align*} T\cos20^\circ=&\,\frac34(392-T\sin20^\circ),\\ T\left(\cos20^\circ+\frac34\sin20^\circ\right)=&\,294,\\ T=&\,\frac{294}{\cos20^\circ+\frac34\sin20^\circ}. \end{align*}

Therefore

\boxed{T=246\text{ N}}\quad(3\text{ s.f.}).