Final answer:
The factor by which the kinetic energy of the system changes after the putty is dropped onto the turntable is 0.523.
Step-by-step explanation:
To find the factor by which the kinetic energy of the system changes after the putty is dropped onto the turntable, we need to compare the initial kinetic energy to the final kinetic energy. The initial kinetic energy is given by the formula K₁ = (1/2)Iω₁², where I is the moment of inertia and ω₁ is the initial angular velocity. Since the turntable is initially spinning freely, ω₁ can be calculated using the formula ω₁ = (2πf₁)/60, where f₁ is the initial frequency in revolutions per minute. Substituting the given values, we have ω₁ = (2π(25))/60 = 2.617 rad/s. Substituting this value and the given moment of inertia into the formula for initial kinetic energy, we have K₁ = (1/2)(3.0 × 10^-2 kg×m²)(2.617 rad/s)² = 0.086 J.
The final kinetic energy is given by the formula K₂ = (1/2)Iω₂², where ω₂ is the final angular velocity. When the putty is dropped onto the turntable, it sticks at a point 0.10 m from the center and causes the moment of inertia to change. The new moment of inertia can be found using the parallel axis theorem, which states that I = I_cm + Md², where I_cm is the moment of inertia about the center of mass, M is the mass of the object, and d is the distance between the center of mass and the new axis of rotation. Since the putty is sticking at a point 0.10 m from the center, the new moment of inertia is given by I = I_cm + Md² = (3.0 × 10^-2 kg×m²) + (0.3 kg)(0.10 m)² = 0.033 kg×m². Now we can calculate ω₂ using the formula ω₂ = (2πf₂)/60, where f₂ is the final frequency in revolutions per minute. The final frequency is given as 25 rev/min, so ω₂ = (2π(25))/60 = 2.617 rad/s.
Substituting the values of the new moment of inertia and the final angular velocity into the formula for final kinetic energy, we have K₂ = (1/2)(0.033 kg×m²)(2.617 rad/s)² = 0.045 J.
Finally, we can find the factor by which the kinetic energy changes by dividing the final kinetic energy by the initial kinetic energy: factor = K₂/K₁ = 0.045 J / 0.086 J = 0.523.