题目

Problem

A small block of mass $5\,\mathrm{kg}$ lies at rest on a rough horizontal plane.

The coefficient of friction between the block and the plane is $\frac{3}{7}$ .

A force of magnitude $P$ newtons is applied to the block in a direction which makes an angle of $30^\circ$ with the plane, as shown in Figure 1.

The block is modelled as a particle.

Given that $P=14$ ,

(a) find the magnitude of the frictional force exerted on the block by the plane and describe what happens to the block, justifying your answer.

(6)

The value of $P$ is now changed so that the block is on the point of slipping along the plane.

(b) Find the value of $P$ .

(6)

题目中文翻译

一个质量为 $5\,\mathrm{kg}$ 的小物块静止在粗糙水平面上。

物块与平面之间的摩擦系数为 $\frac{3}{7}$ 。

如图 1 所示，一个大小为 $P$ 牛顿的力作用在物块上，方向与平面成 $30^\circ$ 角。

将物块视为质点。

已知 $P=14$ ，

(a) 求平面对物块施加的摩擦力大小，并描述物块会如何运动或保持何种状态，说明答案的理由。

(6)

现改变 $P$ 的值，使物块恰好要沿平面滑动。

(b) 求 $P$ 的值。

(6)

解答

(a)

解法一

思路

展开

先求支持力，再算最大静摩擦力。实际静摩擦力只需平衡水平外力，若所需大小小于最大值，物块仍静止。

答题过程

展开

Resolving vertically, the normal reaction is

R=5g-14\sin30^\circ=49-7=42\ \mathrm N.

The maximum available static friction is

F_{\max}=\mu R=\frac37(42)=18\ \mathrm N.

The horizontal component of the applied force is

14\cos30^\circ=7\sqrt3=12.124\ldots\ \mathrm N.

Since $7\sqrt3<18$ , static friction can balance this horizontal force. The block therefore remains at rest, and the actual frictional force has magnitude

\boxed{F=7\sqrt3\ \mathrm N\approx12.1\ \mathrm N}

acting opposite to the horizontal component of the applied force.

(b)

解法一

思路

展开

临界滑动时才有 $F=\mu R$ 。随着拉力改变，支持力也改变，所以必须重新把支持力表示成 $P$ 的函数。

答题过程

展开

Let the new normal reaction be $S$ . Vertical equilibrium gives

S+P\sin30^\circ=5g,\qquad S=49-\frac P2.

At limiting equilibrium, horizontal equilibrium gives

P\cos30^\circ=\frac37S.

Substituting for $S$ ,

\begin{align*} \frac{\sqrt3}{2}P=&\,\frac37\left(49-\frac P2\right),\\ 7\sqrt3P=&\,294-3P,\\ P=&\,\frac{294}{7\sqrt3+3}=\boxed{19.4\ \mathrm N}\quad\text{(3 s.f.)}. \end{align*}

At this value $S>0$ , so contact with the plane is maintained.

M1 2022 June Q4

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程