A printing head has a constant linear printing speed of 1.80 m/s. The printing head spins in a circle as it prints a spiral circular pattern on a piece of fabric. The inner radius of the pattern is 22 cm while the outer radius is 145 cm. If the printer takes 42.0 minutes to complete the pattern, what is the length of the printed pattern when stretched in a straight line? 

A satellite uses a circular antenna array that rotates to maintain a constant data transmission rate to a ground station. The antenna array has a maximum radius of 2.0 meters. For optimal data transmission, the edge of the array must move at a constant linear speed of 5.0 m/s. The rotation speed can be adjusted as the active transmission elements move from an initial radius of 0.55 meters to the full extent of the array. Determine the rotational frequencies (in units of rpm) when the active elements are at the initial and maximum radii.

In a high-tech laboratory, a circular experimental platform rotates at a speed of 7300 rpm. An observation device is fixed 4.00 cm away from the center of the platform. What is the linear speed of the point beneath the observation device?

A toy car of mass 250 g is rolling on a circular vertical loop of diameter 1.0 m. The coefficient of kinetic friction between the car and the loop is 0.45. At the top of the loop, the speed of the car is 0.25 m/s. Determine the magnitude of the tangential acceleration at the top of the loop.

An 850 g toy sled glides in a circle on a bench (µ_k = 0.4) with the help of a 1.50 m long thread. A fan fitted on the toy thrusts air creating a 6.0 N force perpendicularly to the thread. The thread can withstand a force of 90 N before breaking. Starting from rest, determine the number of circles completed by the toy sled before the thread snaps.

A bicyclist maintains a constant speed of 20 km/h while cycling along a circular path with a constant radius of 25 meters in a park. Sketch and label the total acceleration, radial acceleration, and tangential acceleration vectors for the bicyclist.

A sphere initially at rest is spun in a circle at an angular acceleration α. The sphere's acceleration vector changes as a function of time. Determine the time when the angle between the radius and acceleration vector is θ degrees. Express t in terms of α and θ.

Initially resting on a frictionless ice rink, a solid uniform steel disk with a mass of 25.0 kg and a radius of 60.0 cm is pulled by a rope wound around its circumference. A constant horizontal force of 30.0 N is applied to the rope, pulling it to the right. In which direction does the center of mass (CM) of the disk move?