A physics student launches a ball with an initial angular speed of 85.0 rpm on a flat circular path. The radius of the path is 35 cm. The ball is subject to rolling friction. Calculate the time it will take the ball to come to rest. The coefficient of rolling friction is 0.25.

The speed of a particle rotating in a circle of radius 0.5 m is increased from rest at a constant rate of 0.25 m/s². At time t, the magnitude of the centripetal acceleration equals the magnitude of the tangential acceleration. Determine the time t.

A racing kart accelerates from 15 m/s to 20 m/s as it rounds a tight corner on a flat racetrack. The corner is part of a circular arc with a constant radius of 30 meters. Sketch and label the total acceleration, radial acceleration, and tangential acceleration vectors for the kart at the moment it is halfway through the corner.

A truck decreases its speed from 70 km/h to 50 km/h as it takes a curved exit ramp off a highway. The ramp curves with a constant radius of 100 meters. Sketch and label the total acceleration, radial acceleration, and tangential acceleration vectors for the truck as it slows down on the exit ramp.

Imagine a drone with mass m flying in a circle of radius r. Its speed changes over time according to $a_{tan}$ = a + bt², where a and b are constants and t is time in seconds. Starting with an initial speed v₀ at t=0, find the formulas for the tangential force ($F_{tan}$) and radial force ($F_R$) acting on the drone for any time t greater than 0.