A block (mass =2.0 kg ) is hanging from a massless cord that...

A block (mass $=2.0 \mathrm{kg}$ ) is hanging from a massless cord that is wrapped around a pulley (moment of inertia $=1.1 \times 10^{-3} \mathrm{kg} \cdot \mathrm{m}^{2}$ ), as the drawing shows. Initially the pulley is prevented from rotating and the block is stationary. Then, the pulley is allowed to rotate as the block falls. The cord does not slip relative to the pulley as the block falls. Assume that the radius of the cord around the pulley remains constant at a value of $0.040 \mathrm{m}$ during the block's descent. Find the angular acceleration of the pulley and the tension in the cord.

Key Concepts

Newton's Second Law for Translational Motion

This principle states that the net force acting on an object is equal to its mass multiplied by its acceleration (F = ma). In the context of the falling block, it is used to set up an equation where gravitational force minus the tension in the cord produces the net force acting on the block, leading to its acceleration.

Rotational Dynamics and Torque

Rotational dynamics involves analyzing how torques affect rotational motion. The torque acting on the pulley, provided by the tension in the cord, is equal to the product of the pulley’s moment of inertia and its angular acceleration (? = I?). This relation is essential to connect the effect of the tension on the block with the angular acceleration of the pulley.

Kinematic Relationship between Linear and Angular Motion

When a cord does not slip on a pulley, the linear acceleration of the block is directly related to the angular acceleration of the pulley by the relation a = r?, where r is the radius of the pulley. This relationship links the translational motion of the block to the rotational motion of the pulley.

Moment of Inertia

The moment of inertia measures an object's resistance to changes in its rotational motion. For the pulley, it quantifies how much torque is needed to achieve a given angular acceleration. This concept is crucial for relating the applied torque (via the cord tension) to the angular acceleration using the formula ? = I?.

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