Converting Between Linear & Rotational: Videos & Practice Problems

A cyclone used for roof ventilation has a diameter of 60.0 cm. Suppose the cyclone has a steady angular acceleration of 0.200 rev/s². At t= 0, the cyclone has an angular velocity of 0.450 rev/s. At t = 0.800s, determine the tangential speed of a point at the edge of the cyclone.

A sewing machine has a speed of 4600 stitches per minute. A stitch is 2.00 mm, giving a constant linear sewing speed of 9200 mm/min. The sewing machine is performing embroidery on rotating fabric. The pattern of the embroidery is a spiral circle whose radius increases outwards. The inner radius of the pattern is 0.80 cm while the outer radius is 10.0 cm. If the embroidery is completed in 25.0 minutes, determine the average angular acceleration of the fabric for the 25.0-minute period. Take the direction of the angular velocity as the positive direction. 

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 cogwheel has a diameter of 25.4 cm. Suppose a cogwheel has a steady angular acceleration of 0.320 rev/s². At t = 0, it has an angular velocity of 0.250 rev/s. Determine the acceleration magnitude of a point at the edge cog of the cogwheel when t = 0.350 s.

A polymer drop is placed on a rotating disk with an angular speed of ω, and it experiences a centripetal acceleration of 4.5 m/s². The disk rotational speed is increased to 3ω. Calculate the drop's centripetal acceleration.

A circular tabletop has a radius of 1 m. You want to print a spiral decoration whose radius increases outward by spinning the top. The printing head has a constant linear printing speed of 50 mm/s. The decoration starts at an inner radius of 0.10m and ends at an outer radius of 0.90 m. Determine the angular speed of the tabletop when printing the innermost and outermost parts of the decoration.

A parked vehicle drives off giving the tires a uniform angular acceleration of 8.00 rad/s². If each tire has a radius of 20.5 cm, what is the radial acceleration at a point on the threaded part of the tire when it completes 4 revolutions using the equation a_rad = v²/r?

A thin glass container is placed 7.5 cm from the axis of rotation of a spin coater device. The spin coater rotates at 2500 rpm. Calculate the container's centripetal acceleration.

A sphere attached to a rod is rotated, starting from rest, in a circular path of diameter d with a constant angular acceleration α. The sphere undergoes an angular displacement of  θ₂-θ₁. Write the expressions of the sphere's linear velocity and radial acceleration in terms of d, α, and θ₂-θ₁.

Shafts are fitted with disks to create useful devices for storing rotational kinetic energy. The disks increase their rotational speeds when extra energy is available and decrease their rotational speeds when required to deliver energy. A shaft fitted with tungsten carbide ceramic bearings spins at 10500 rpm. Calculate the speed at a point located on the edge of a 22 cm diameter disk that is fitted to the rotating shaft.

Determine the speed of a point at the top of a skateboard's wheel that has a diameter of 68.0 cm after 2.76 s has passed, when the skateboarder accelerates from rest at a rate of 1.50 m/s². Please note that the wheel's lowest point remains stationary and in contact with the ground at any moment in time.

A laboratory mixer spins a sample located 8.1 cm from the rotation axis to undergo 200,000 g's acceleration. Determine how fast (in rpm) must it rotate?

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.

To stabilize its orientation for a communication link, a satellite initiates a maneuver to rotate about its axis. Initially stationary, the satellite accelerates to achieve a rotational speed of 2.0 revolutions per minute in just 10.0 seconds. Assume the satellite is cylindrical, with a radius of 1.0 meters, rotating about its longitudinal axis. Calculate both the radial and tangential components of the linear acceleration of a point on the outer surface of the satellite 5.0 seconds after it begins this maneuver.

A discus thrower increases the speed of a 2.0 kg discus from zero to 24 m/s while making 2.5 complete rotations. Assuming the angular velocity increases at a constant rate and that the discus moves in a flat circle with a radius of 2.0 meters, determine the tangential acceleration, disregarding air resistance and gravity.

A robotic arm in a manufacturing facility is designed to transport materials by rapidly spinning them in a circular path before releasing them onto a conveyor belt. The arm picks up a material packet of mass 0.8 kg and accelerates it from rest, completing 4.0 revolutions before releasing the packet at a speed of 20.0 m/s. The circular path has a radius of 1.0 m. Assuming a uniform rate of increase in angular velocity, calculate the centripetal acceleration of the material packet just before its release. Ignore the effects of air resistance.

An athlete prepares to throw a discus by spinning in place, holding the discus at the end of an arm's length, 1.2 meters from the pivot point. The discus has a mass of 2.0 kg and is accelerated from rest to a final speed of 25 m/s after completing 3.0 full revolutions. Assuming the rate of angular acceleration is uniform, calculate the net force being exerted on the discus by the athlete just before its release. Ignore the effects of gravity for this calculation.

In an automated assembly line, a robotic arm swiftly moves a component of mass 0.50 kg in a circular path with a radius of 0.75 m. The component is accelerated from rest and completes 5.0 full revolutions before being released at a speed of 18.0 m/s onto a designated spot. Assuming the angular acceleration is uniform throughout the motion, calculate the angle of the force exerted by the robotic arm on the component with respect to the radius of the circular path just before release. Ignore the effects of gravity in this scenario.

A satellite is in a stable circular, counterclockwise orbit around the Earth, moving with a constant speed. The satellite's position in the Earth-centered inertial frame, projected onto the equatorial plane, is given by the vector equation:

$\vec{r}=R\cos (\omega t)\ \hat{i}+R\sin (\omega t)\ \hat{j}$,

where $R$ is the radius of the orbit, and $ω = v/R$ is the angular velocity with respect to Earth. Assuming the orbit is counterclockwise, determine the velocity $\vec{v}$ and angular velocity $\vec{\omega }$ of the satellite. Is $\vec{v}$ = $\vec{\omega }$ $\times$ $\vec{r}$?

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?

In a supercomputer's memory disk rotating at 7400 rpm, how many bits can the laser writer store per second if each bit requires 0.60 μm of space and the writer is 5.00 cm from the center?

A uniformly rotating circular saw blade has a diameter of 45 cm. Starting at an initial speed of 150 rpm, its angular velocity will increase to a final speed of 350 rpm in 6.0 s. Considering a point on the blade's edge, calculate the radial and tangential components of linear acceleration, 4.0 seconds after it starts accelerating.