Question

2 a solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. derive an expression for its linear acceleration.

Explanation:

Step1: 受力分析与牛顿第二定律

设摩擦力为\(f\)，根据牛顿第二定律\(Mg\sin\theta - f = Ma\)。

Step2: 转动定律

对于球体，转动惯量\(I=\frac{2}{5}MR^{2}\)，根据转动定律\(fR = I\alpha\)，又因为无滑滚动\(a = R\alpha\)，则\(f=\frac{I a}{R^{2}}=\frac{2}{5}Ma\)。

Step3: 联立求解

将\(f=\frac{2}{5}Ma\)代入\(Mg\sin\theta - f = Ma\)，得到\(Mg\sin\theta-\frac{2}{5}Ma = Ma\)，整理可得\(a=\frac{5}{7}g\sin\theta\)。

Answer:

\(a=\frac{5}{7}g\sin\theta\)