Final answer:
To find the puck's speed at the smaller radius, we can use the principle of conservation of angular momentum.
Step-by-step explanation:
To find the puck's speed at the smaller radius, we can use the principle of conservation of angular momentum. The formula for angular momentum is given by L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. Since the cord is being pulled and the radius is decreasing, the moment of inertia changes, but the angular momentum remains constant. First, we need to calculate the moment of inertia at both radii.
The moment of inertia of a circle is given by I = (1/2)mr^2, where m is the mass and r is the radius. At the initial radius, the moment of inertia is Ii = (1/2)mri^2. At the smaller radius, the moment of inertia is If = (1/2)mrf^2. Since the mass remains constant, we can write Ii/Ii = rf^2/ri^2.
Solving for rf, we get rf = sqrt(If/Ii) * ri. Now we can calculate the angular velocity at the smaller radius using the formula L = Iiωi = Ifωf. Rearranging the equation, we get ωf = Iiωi/If. Plugging in the values, we have ωf = ωi * If/Ii = ωi * (1/2) * (If/Ii).
Finally, we can calculate the linear speed of the puck using the formula v = ω * r. Plugging in the value for ωf and the smaller radius, we get v = ωf * r = (ωi * (1/2) * (If/Ii)) * rf.