Rotational Velocity & Acceleration Practice Problems

The angular position of a small sphere welded on a spinner wheel is given in the graph below. Calculate the angular velocity at t = 3 s and t =10 s

The equation ω_z(t) = B - Dt² gives the angular velocity of a rotating particle, where B = 3.80 rad/s and D = 0.940 rad/s³. Determine the average angular acceleration for the interval t = 0 - 2.50 s and instantaneous angular acceleration when t = 2.50 s. α_av-z = ? α_z = ?

The equation ω_z(t) = C - Dt² gives the angular velocity of a rotating particle, where C = 4.20 rad/s and D = 0.250 rad/s³. Express its angular accleration in the form α(t) = ? 

A decoration on a rotating disk has angular velocity given by ω_z(t) = C + Dt² where C and D are constants and t is in seconds. The numerical values of C and D are 3.25 and 0.850 respectively. Determine the decoration's angular acceleration at t = 0 s and t = 3.60 s. 

A sphere follows a circular trajectory such that its angular position follows θ(t) = X + Yt - Zt³, where θ is measured in radians, t is measured in seconds, and X, Y, and Z are constants. At t = 0, the angular position of the X is θ = π/3 while the angular velocity is 1.30 rad/s. In another instance, t = 1.80 s, the sphere has an angular acceleration of 0.900 rad/s². At the moment where angular accleration is 2.80 rad/s², determine the values of angular velocity and angular position.

A toy Ferris wheel spins such that its angular position follows the equation θ(t) = X + Yt - Zt³, where θ is measured in radians, t is measured in seconds, and X, Y, and Z are constants. At t = 0, the angular position of the wheel is θ = π/5 while the angular velocity is 1.60 rad/s. At another instance of time, t = 1.40 s, the wheel has an angular acceleration of 1.30 rad/s². Determine the angular acceleration of the wheel at the moment when θ = π/5 rad.

A toy Ferris wheel spins such that its angular position follows the equation θ(t) = X + Yt - Zt³, where θ is measured in radians, t is measured in seconds, and X, Y, and Z are constants. At t = 0, the angular position of the wheel is θ = π/2 while the angular velocity is 1.70 rad/s. At another instance of time, t = 2.20 s, the wheel has an angular acceleration of 0.875 rad/s². Determine the values of  X, Y, and Z and indicate their units. 

A wind turbine of diameter 1200 mm spins at an angular velocity ω = [8.0 - (1/4)t²] rad/s, where t is measured in seconds. Find the turbine's angular acceleration at t = 6.0 s.

A sprocket has a circumference of 56.55 cm measured along the teeth tips. The angular velocity of the sprocket is modeled by ω = [6.0 + (1/6)t²] rad/s , where the units of t is s. Determine the tangential acceleration at the tip of a tooth on the sprocket when t = 9.0 s.

The position of a hole engraved on a rotating disk obeys the position equation θ = (8.0 rad/s⁴)t⁴. Determine the angular acceleration of the hole after 15 rotations of the disk.

A shaft has an angular velocity function ω  = (25 - t²) rad/s, where t is measured in seconds. Calculate Δθ for the shaft between t = 0s and the time when the direction of the spinning shaft is reversed.

A blender motor accelerates evenly from rest to 18500 rpm in 2.0s, maintaining that speed for 8.0s, and finally decelerates evenly to a complete stop in 2.50 s. Determine the number of revolutions made.

A large disk with a diameter of 3.0 m spins with a period of 2.0 s. The disk is slowed down to a complete stop in 15 s. Calculate the number of turns made by the disk before stopping.

A wheel has a decoration mounted 20 cm from the axis of the 700 mm wheel. An observer notices that the wheel makes 5 turns every second. Calculate the speed (in m/s) and net acceleration of the decoration.

A circular disk of diameter 15cm is attached to an electric motor. The disk has a coupling screw located 7cm from the center of the disk. The disk is accelerated at 1400 rad/s² in a time t = 1.5s and allowed to spin at uniform angular velocity for 1s. Determine the number of rotations made by the disk.

A truck tire has a diameter of 1130 mm. The truck is driven at 62 km/h. Find the speed of a pebble stuck in the tire threads when it reaches the top of the tyre. Assume the pebble is located on the edge of the tyre.

A vehicle is fitted with a tubeless, 1160 mm diameter tire. The vehicle is driven at a speed of 80 miles per hour. A repair strip used in puncture repair is visible at the outer edge (threaded area) of the tire. Determine the speed of the visible section of the repair strip when it is located at the bottom of the tire.

A periodic varying force acts on a shaft. The graph below shows the angular velocity of the shaft. Find the angular acceleration of the crankshaft at t = 8 s.

The graph below shows an electronically generated line of the best fit using measured angular velocities for a wheel. The velocity is measured every 0.4 s. Find the wheel's initial angular velocity when t = 0 s.

A circular disk spinning in the vertical plane is used to launch masses along a vertical tangent. The disk has a diameter of 0.50 m and the marble to be launched has a mass of 14.5 g. The disk is initially at rest, then it is accelerated at 750 rad/s². If the marble is launched after 0.5 revolutions, calculate the maximum height attained above the launch point.

A shaft acted on by multiple torques has angular velocity given by ω = (5 - 0.2t²), t being in seconds. When will the shaft experience reversal in its direction of rotation?

Disks are efficient devices for storing rotational kinetic energy. Their rotational energy increases when surplus energy is available and will lose rotational kinetic energy when the disks do work on a load. Tungsten carbide ceramic bearings allow shafts to spin at speeds as high as 10200 rpm. If a 22 cm disk loses 35% of its rotational velocity in 20 s as it does work on a load, determine the magnitude of the disk's angular acceleration. Take the angular acceleration to be constant. 

A ceiling fan starts to accelerate with an angular acceleration α = 6.7t² - 3.0t from rest. Here, α is given in rads/s² and time t is in seconds. Derive an expression for the angular displacement Φ as a function of time given that, when t = 0, Φ = 0, ω = 0.

A wire is being unwounded from a spool initially at rest as shown in the figure. If the tension T in the wire is given by T = 5.00t - 0.40t² (newtons) where t is the time in seconds, and the moment of inertia of the spool is 0.19 kg.m², determine the linear speed of a point on the outer edge of the spool after 7.0 s. (Assume that friction is negligible.)

A spinning top is rotating on a horizontal surface. When viewed from above, the top spins clockwise at an angular velocity ω₁ = 60.0 rad/s about its axis, and the surface is rotating anticlockwise with an angular velocity ω₂ = 45.0 rad/s about a vertical axis. What are the directions of ω₁ and ω₂ at a given instant?

A new bicycle wheel has been developed and put on display. It is mounted on a rotating display platform. The wheel spins at 60.0 rad/s about its axle, while the platform rotates at 40.0 rad/s around a vertical axis. What is the resultant angular velocity of the wheel as observed by someone standing away from the platform? Find the magnitude and direction of the angular velocity.

A new bicycle wheel has been developed and put on display. It is mounted on a rotating display platform. The wheel spins at 60.0 rad/s about its axle, while the platform rotates at 40.0 rad/s around a vertical axis. What is the magnitude and direction of the angular acceleration of the spinning wheel at a given instant? Assume that the Z axis is vertically upward and that, at this moment, the X axis points to the right along the direction of rotation.