question_answer

The mass of a solid sphere is 2 kg and radius is 10 cm. Sphere, tied to one end of 1 m long string, is rotated at a rate of 10rad/sec. The rotational kinetic energy of sphere is :

A) 121.4 J                                  

B) 185J

C) 15.2 J                                    

D) 22.0 J

Correct Answer: A

Solution :

Moment of inertia about diameter is \[{{I}_{G}}=\frac{2}{5}M{{r}^{2}}\] Distance of diameter from rotational axis \[d=1\text{ }m+10\text{ }cm=1.1\text{ }m\] From the theorem of parallel axis \[I={{I}_{G}}+M{{r}^{2}}\]                 \[=\frac{2}{5}M{{r}^{2}}+M{{d}^{2}}\]                 \[=\frac{2}{5}\times 2\times {{(0.1)}^{2}}+2\times {{(1.1)}^{2}}\] \[=0.008+2.42\] \[=2.428\text{ }kg-{{m}^{3}}\] Therefore, rotational kinetic energy of sphere \[K=\frac{1}{2}I{{\omega }^{2}}\] \[=\frac{1}{2}\times 2.428\times {{(10)}^{2}}\] \[=121.4\text{ }J\]