A 245 g toy car is placed on a narrow 70-cm-diameter track with wheel grooves that keep the car going in a circle. The 1.0 kg track is free to turn on a frictionless, vertical axis. The spokes have negligible mass. After the car's switch is turned on, it soon reaches a steady speed of 0.50 m/s relative to the track. What then is the track's angular velocity, in rpm? (State the magnitude of the angular velocity.)

To find the track's angular velocity, we need to use the principle of conservation of angular momentum.

The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Before the switch is turned on, the toy car and the track are at rest, so the initial angular momentum is zero. After the car's switch is turned on and it starts moving, the system will have a final angular momentum.

The moment of inertia of the system is the sum of the moment of inertia of the car and the moment of inertia of the track. The moment of inertia of a point mass rotating around an axis is given by I = mr^2, where m is the mass of the object and r is the radius.

Given that the car has a mass of 245 g (0.245 kg) and the track has a diameter of 70 cm (0.7 m), we can calculate the moment of inertia of the car as I_car = m_car * r_car^2 = 0.245 * (0.35)^2 = 0.0308125 kg * m^2.

The moment of inertia of the track can be calculated as the moment of inertia of a disc rotating around its central axis: I_track = (1/2) * m_track * r_track^2 = (1/2) * 1 * (0.35)^2 = 0.030625 kg * m^2.

The total moment of inertia of the system is I_total = I_car + I_track = 0.0308125 + 0.030625 = 0.0614375 kg * m^2.

Next, we need to calculate the final angular momentum of the system. The car's final speed relative to the track is given as 0.50 m/s. Since the car is moving in a circle, its velocity is tangent to the circle at any point. Therefore, the velocity of the car is the same as the tangential velocity of a point on the track.

The tangential velocity can be calculated using the formula v = ω * r, where v is the tangential velocity, ω is the angular velocity, and r is the radius of the circle.

Rearranging the equation, we get ω = v / r = 0.50 / 0.35 = 1.43 rad/s.

Finally, we need to convert the angular velocity from radians per second to revolutions per minute (rpm). There are 2π radians in one revolution, and 60 seconds in one minute.

Thus, the angular velocity in rpm is ω_rpm = (1.43 * 60) / (2π) = 13.66 rpm.

So, the track's angular velocity is approximately 13.66 rpm.