题目

Problem

A broom is being used to sweep a rough horizontal floor. The handle of the broom makes a constant angle of $40^\circ$ with the horizontal, as shown in Figure 1. The broom head is modelled as a particle of mass $0.5\,\mathrm{kg}$ and the handle of the broom is modelled as a light rod. The coefficient of friction between the broom head and the floor is $\frac{1}{4}$ . The broom head is pushed along the floor in a straight line at constant speed. Find the magnitude of the force that is being applied along the handle of the broom to the broom head.

(6)

题目中文翻译

如图 1 所示，用一把扫帚清扫粗糙水平地面，扫帚柄与水平方向始终成 $40^\circ$ 角。将扫帚头视为质量为 $0.5\,\mathrm{kg}$ 的质点，将扫帚柄视为轻杆。扫帚头与地面之间的摩擦系数为 $\frac{1}{4}$ 。扫帚头被推动，沿地面做匀速直线运动。求沿扫帚柄施加在扫帚头上的力的大小。

(6)

解答

解法一

思路

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推力沿扫帚柄斜向下，因此会增大支持力。匀速意味着水平合力为零，但支持力不能直接写成重力。

答题过程

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Let the applied force be $P$ , the normal reaction be $R$ and the friction be $F$ . The force $P$ has a horizontal forward component $P\cos40^\circ$ and a downward component $P\sin40^\circ$ .

Vertical equilibrium gives

R=0.5g+P\sin40^\circ.

Since the broom head is sliding, $F=\mu R=R/4$ . Constant speed gives zero horizontal resultant, so

P\cos40^\circ=F=\frac14R.

Eliminating $R$ ,

\begin{align*} 4P\cos40^\circ=&\,0.5g+P\sin40^\circ,\\[4mm] P(4\cos40^\circ-\sin40^\circ)=&\,0.5g. \end{align*}

Using $g=9.8\,\mathrm{m\,s^{-2}}$ ,

P=\frac{4.9}{4\cos40^\circ-\sin40^\circ} =\boxed{2.02\,\mathrm{N}}\quad\text{(3 s.f.)}.

M1 2019 June Q2

题目

解答

解法一

思路

答题过程