The figures I, II, and III are parts of a sequence. Which one of the following options comes next in the sequence at IV?
The pattern involves the shaded area rotating clockwise by a quarter turn each step.
Step 1: In Figure I, the top left quarter is shaded.
Step 2: In Figure II, the bottom left quarter is shaded.
Step 3: In Figure III, the bottom right quarter is shaded.
Step 4: Following this pattern, the next figure should have the top right quarter shaded, but as per the provided options, the closest match under a clockwise movement is the left quarter shaded for a continuation of the sequence in a new cycle.
Therefore, the correct answer, aligning with a continuous cycle of the sequence, is that the left quarter should be shaded in the next figure.
Which letter replaces the question mark? A, D, G, J, M, ?
A wheel of mass \( 4M \) and radius \( R \) is made of a thin uniform distribution of mass \( 3M \) at the rim and a point mass \( M \) at the center. The spokes of the wheel are massless. The center of mass of the wheel is connected to a horizontal massless rod of length \( 2R \), with one end fixed at \( O \), as shown in the figure. The wheel rolls without slipping on horizontal ground with angular speed \( \Omega \). If \( \vec{L} \) is the total angular momentum of the wheel about \( O \), then the magnitude \( \left| \frac{d\vec{L}}{dt} \right| = N(MR^2 \Omega^2) \). The value of \( N \) (in integer) is: