A hoop of mass \( M \) and radius \( R \) rolls without slipping along a straight line on a horizontal surface as shown in the figure. A point mass \( m \) slides without friction along the inner surface of the hoop, performing small oscillations about the mean position. The number of degrees of freedom of the system (in integer) is \(\underline{\hspace{2cm}}\).
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Show Hint
Correct Answer: 2
Solution and Explanation
2. The point mass \( m \), which is sliding along the inner surface of the hoop.
- The hoop has one degree of freedom due to its motion along the straight line.
- The point mass \( m \) performs small oscillations about the mean position, which contributes one additional degree of freedom.
Thus, the total number of degrees of freedom for the system is the sum of the degrees of freedom of both parts: \[ \text{Degrees of freedom} = 1 \ (\text{hoop}) + 1 \ (\text{point mass}) = 2. \] Therefore, the number of degrees of freedom of the system is 2.
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