2 a solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. derive an…
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.
Answer
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)