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3.27 m

3.23 rad/s

Explanation:

There are external forces on the sphere, so momentum is not conserved, but energy is conserved.

Initial KE + Initial RE = Final PE

½ Mv² + ½ Iω² = Mgh

½ Mv² + ½ (⅖ MR²) ω² = Mgh

½ Mv² + ⅕ MR²ω² = Mgh

5v² + 2R²ω² = 10gh

For rolling without slipping, ωR = v.

5v² + 2v² = 10gh

7v² = 10gh

h = 0.7 v² / g

h = 0.7 (3.90 m/s)² / (10 m/s²)

h = 1.0647 m

Use trig to find the distance along the ramp:

sin 19.0° = h / d

d = h / sin 19.0°

d = 3.27 m

The clay sticks to the turntable, so this is an inelastic collision.  Kinetic energy is not conserved, but angular momentum is conserved.

L₀ = L

I₀ ω₀ = I ω

(½ MR²) ω₀ = (½ MR² + mr²) ω

MR²ω₀ = (MR² + 2mr²) ω

ω = MR²ω₀ / (MR² + 2mr²)

ω = (0.190) (0.200)² (4.00) / (0.190 (0.200)² + 2 (0.040) (0.150)²)

ω = 3.23 rad/s