本题目来源于试卷: 2012美国US F=MA物理竞赛，类别为 美国F=MA物理竞赛
[单选题]
Four objects are placed at rest at the top of an inclined plane and allowed to 61vh,c5w gvhu*dqan0 rh6wv0avc5 gnhqd ,1u* )v(v oyg7 r:huoll without slipping to the bottom in the absence of rolling resistance and arvvyo h u()g7:ir resistance.
* Object A is a solid brass ball of diameter $d$.
* Object B is a solid brass ball of diameter $2d$.
* Object C is a hollow brass sphere of diameter $d$.
* Object D is a solid aluminum ball of diameter $d$. (Aluminum is less dense than brass.)
The balls are placed so that their centers of mass all travel the same distance. In each case, the time of motion $T$ is measured. Which of the following statements is correct?
A. $T_B > T_C > T_A = T_D$
B. $T_A = T_B = T_C > T_D$
C. $T_B > T_A = T_C = T_D$
D. $T_C > T_A = T_B = T_D$
E. $T_A = T_B = T_C = T_D$
参考答案: D
本题详细解析:
The acceleration of an object rolli2yx z(uodm.e)s,6 lco ng without slipping down an incline is $a = \frac{g \sin \theta}{1 + I/(mR^2)}$.
For a solid sphere (Objects A, B, D), the moment of inertia is $I = \frac{2}{5}mR^2$.
$a_{solid} = \frac{g \sin \theta}{1 + (\frac{2}{5}mR^2)/(mR^2)} = \frac{g \sin \theta}{1 + 2/5} = \frac{5}{7}g \sin \theta$.
This acceleration is independent of mass $m$, radius $R$, or density. Therefore, $a_A = a_B = a_D$.
For a hollow sphere (Object C), $I = \frac{2}{3}mR^2$ (assuming a thin shell).
$a_{hollow} = \frac{g \sin \theta}{1 + (\frac{2}{3}mR^2)/(mR^2)} = \frac{g \sin \theta}{1 + 2/3} = \frac{3}{5}g \sin \theta$.
Since $\frac{3}{5} \approx 0.6$ and $\frac{5}{7} \approx 0.714$, we have $a_{hollow} < a_{solid}$.
Since the distance $L$ is constant, $L = \frac{1}{2}aT^2$, which means $T = \sqrt{2L/a}$.
A smaller acceleration $a$ results in a larger time $T$.
Therefore, $T_C > T_A = T_B = T_D$.
| 
