To determine the angular velocity of the wheel after the cord is unwound by 1 meter, we start by calculating the work done by the force applied on the cord.

Work Done by Force (W_F):

$ W_F = 20 \, \text{N} \times 1 \, \text{m} = 20 \, \text{J} $

Relationship with Kinetic Energy:

The work done (W_F) is converted into the change in kinetic energy of the wheel. Therefore, we have:

$ \Delta \text{KE} = 20 \, \text{J} = \frac{1}{2} I \omega^2 $

where $ I $ is the moment of inertia of the wheel, and $ \omega $ is the angular velocity.

Determine the Moment of Inertia (I):

Since the wheel has mass $ M = 10 \, \text{kg} $ and radius $ R = 10 \, \text{cm} = 0.1 \, \text{m} $, and knowing it's a solid disk:

$ I = MR^2 = 10 \times 0.1^2 = 0.1 \, \text{kg} \cdot \text{m}^2 $

Solve for Angular Velocity ($\omega$):

Apply the equation for kinetic energy:

$ 20 = \frac{1}{2} \times 0.1 \times \omega^2 $

Solving for $\omega$, we get:

$ 20 = 0.05 \omega^2 $

$ \omega^2 = \frac{20}{0.05} = 400 $

$ \omega = \sqrt{400} = 20 \, \text{rad/s} $

Therefore, the angular velocity of the wheel after the cord is unwound by 1 meter is $ 20 \, \text{rad/s} $.