Final answer:
The mass of the wheel is 36.18 kg.
Step-by-step explanation:
To find the mass of the wheel, we first need to calculate the acceleration of the object. We can use the equation of motion: s = ut + 0.5at^2 where s is the distance, u is the initial velocity (which is 0 as the object is released from rest), a is the acceleration, and t is the time. Rearranging the equation, we get: a = 2s/t^2. Substituting the given values, we have: a = 2(3.10 m) / (2.15 s)^2 = 0.664 m/s^2.
Now, we can use the equation F = ma to find the force acting on the object. Since the object is hanging vertically, the force acting on it is its weight, which is given by F = mg. Substituting the values, we have: mg = m(9.8 m/s^2) = 4.40 kg * 9.8 m/s^2 = 43.12 N. Therefore, the force acting on the object is 43.12 N.
Finally, we can find the force exerted by the wheel on the wire. Since the wire is wrapped around the rim of the wheel, the force exerted by the wheel on the wire is also the tension in the wire. This tension is equal to the force acting on the object, so the force exerted by the wheel on the wire is also 43.12 N.
The torque exerted on the wheel can be calculated using the equation τ = Fr, where τ is the torque, F is the force, and r is the radius of the wheel. Substituting the values, we have: τ = (43.12 N)(0.278 m) = 11.97 Nm. Since the wheel rotates without friction, the net torque on the wheel is zero. Therefore, the torque exerted by the object hanging from the wire is equal in magnitude but opposite in direction to the torque exerted by the wheel. Thus, the torque exerted by the object is also 11.97 Nm.
The torque exerted by the object can be calculated using the equation τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Rearranging the equation, we get: α = τ/I. Substituting the values, we have: α = 11.97 Nm / 6.0 kg m² = 1.995 rad/s².
Since the object moves downward, the angular acceleration is in the same direction as the angular velocity (clockwise). The angular velocity can be calculated using the equation of motion for rotational motion: ω = ω₀ + αt, where ω is the final angular velocity, ω₀ is the initial angular velocity (which is 0 as the wheel is released from rest), α is the angular acceleration, and t is the time. Substituting the values, we have: ω = 0 + (1.995 rad/s²)(2.15 s) = 4.294 rad/s.
Since the angular velocity is directly proportional to the rotational speed, we can calculate the rotational speed using the equation v = ωr, where v is the tangential velocity and r is the radius of the wheel. Substituting the values, we have: v = (4.294 rad/s)(0.278 m) = 1.192 m/s
To find the mass of the wheel, we can use the equation m = ΣF / a, where m is the mass, ΣF is the sum of the forces, and a is the acceleration. In this case, the only force is the tension in the wire exerted by the wheel, and the acceleration is the tangential acceleration of the wheel. Substituting the values, we have: m = 43.12 N / (1.192 m/s²) = 36.18 kg. Therefore, the mass of the wheel is 36.18 kg.