A) solid sphere reaches the ground first
B) hollow sphere reaches the ground first
C) both spheres reaches the ground at the same time
D) the time at which the spheres reach the ground cannot be specified by the given data
Correct Answer: A
Solution :
Key Idea: More the acceleration, faster the mass reaches down. The acceleration of a body rolling down a plane inclined at an angle \[\theta \] is \[a=\frac{g\sin \theta }{1+\frac{{{K}^{2}}}{{{R}^{2}}}}\] For solid sphere \[\frac{{{K}^{2}}}{{{R}^{2}}}=\frac{2}{3}\] For hollow sphere\[\frac{{{K}^{2}}}{{{R}^{2}}}=\frac{2}{3}\] For solid sphere\[{{a}_{s}}=\frac{g\sin \theta }{1+\frac{2}{5}}\] \[\Rightarrow \] \[{{a}_{s}}=\frac{5}{7}\sin \theta \] For hollow sphere\[{{a}_{H}}=\frac{g\sin \theta }{1+\frac{2}{3}}\] \[{{a}_{H}}=\frac{3}{5}g\sin \theta \] Since\[{{a}_{S}}>{{a}_{H}}\], hence solid sphere reaches the ground first.
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