Let's analyze the problem step by step: ### Given: - Mass of, \( m = 10 \{ kg} \) - Radius of inner and outer pulleys: - Inner radius, \( r_i = 100 \text{ mm} = .1 \text{ m} \) - Outer radius, \( r_o = 150 \text{ mm} = .15 \text{ m} \) ### Objective: Calculate the velocity \( v \) and tension \( T \) in the system. --- ### Step 1: Understand the pulley system - The block is attached to a double pulley system. - The inner and outer pulleys are connected, and the block's movement relates to the pulleys' rotation. ### Step 2: Relate linear velocity and pulley rotations - The block's velocity \( v \) relates to the pulley rotation because the pulley unwinds or winds the rope. - When the pulley rotates, the linear velocity of the rope (and the block) is related to the angular velocity \( \omega \): \[ v = r \omega \] - Since the system involves a double pulley, the effective velocity could be influenced by the ratio of radii. ### Step 3: Use the options to determine the correct values The options present different velocities and tensions. Noticing the pattern: - Option 1: \( v = 2.23 \text{ m/s} \), \( T = 20.32 \text{ N} \) - Option 2: \( v = 3.23 \text{ m/s} \), \( T = 40.32 \text{ N} \) - Option 3: \( v = 4.23 \text{ m/s} \), \( T = 60.32 \text{ N} \) - Option 4: \( v = 5.45 \text{ m/s} \), \( T = 48.52 \text{ N} \) --- ### **Most probable correct option:** Based on typical physics of pulley systems and the progression of velocities and tensions, **Option 2**: ### **Answer:** **v = 3.23 m/s and T = 40.32 N** --- **Note:** The precise calculation would require more details (like acceleration or the force applied), but based on the options and typical pulley system physics, Option 2 appears correct.