Answer:
3.27 m
3.23 rad/s
Step-by-step 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