Rotational Kinematics Archives • Nerd Notes

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Flywheels-rapidly rotating disks- are widely used in industry for storing energy. Theu are spun up slowly when extra energy is available, then decelerate quickly when needed to supply a boost of energy. A 20-cm-diameter rotor made of advanced materials can spin at 100,000 rmp.

The angular velocity of a process control motor is [katex]w = \left(20 – \frac{1}{2} t^2 \right) \, \{rad/s} [\katex], where t is in seconds.

A 6.0-cm-diameter gear rotates with angular velocity [katex] w= \left(20-\frac {1}{2} t^2 \right) \text {rad}{s}[\katex], where t is in seconds. At t= 4.0 s, what are:

The angular velocity of a rotating disk of radius 20 cm increases from 1 rad/s to 3 rad/s in 0.5 s. What is the linear tangential acceleration of a point on the rim of the disk during this time interval?

A disk of radius 35 cm rotates at a constant angular velocity of 10 rad/s. How fast does a point on the rim of the disk travel (in m/s)?

A disk increases from 2 complete revolutions in 2 seconds to 5 complete revolutions in 2 seconds. What is its average angular acceleration?

A rotating, rigid body makes 10 complete revolutions in 10 seconds. What is its average angular velocity?

Two masses, my = 32 kg and mg = 38 kg, are connected by a rope that hangs over a pulley. The pulley is a uniform cylinder of radius R = 0.311 m and mass 3.1 kg. Initially my is on the ground and mg rests 2.5 m above the ground. If the system is released, use conservation of energy to determine the speed of me just before it strikes the ground. Assume the pulley bearing is frictionless.

How long does it take for a rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of 3.45 rad/s2?

A pulley has an initial angular speed of 12.5 rad/s and a constant angular acceleration of 3.41 rad/s2. Through what angle does the pulley turn in 5.26 s?

When a fan is turned off, its angular speed decreases from 10 rad/s to 6.3 rad/s in 5.0 s. What is the magnitude of the average angular acceleration of the fan?

A string is wound tightly around a fixed pulley having a radius of 5.0 cm. As the string is pulled, the pulley rotates without any slipping of the string. What is the angular speed of the pulley when the string is moving at 5.0 m/s?

A solid sphere, solid cylinder, and a hollow pipe all have equal masses and radii. If the three of them are released simultaneously at the top of an inclined plane and do not slip, which one will reach the bottom first?

A solid sphere of mass 1.5 kg and radius 15 cm rolls without slipping down a 35° incline that is 7.0 m long. Assume it started from rest. The moment of inertia of a sphere is given by I= (2/5)MR^2.

A centrifuge rotor rotating at 9200 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.20 N·m. If the mass of the rotor is 3.10 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest?

A 0.72-m-diameter solid sphere can be rotated about an axis through its center by a torque of 10.8 N·m which accelerates it uniformly from rest through a total of 160 revolutions in 15.0 s. What is the mass of the sphere?

A rotating merry-go-round makes one complete revolution in 4.0 s. What is the linear speed and acceleration of a child seated 1.2 m from the center?

A centrifuge accelerates uniformly from rest to 15,000 rpm in 240 s. Through how many revolutions did it turn in this time?

A wheel 31 cm in diameter accelerates uniformly from 240rpm to 360rpm in 6.8s. How far will a point on the edge of the wheel have traveled in this time?

The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 5.0 rev/s in 8.0 s. At this point, the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 12.0 s. Through how many revolutions does the tub turn during the entire 20-s interval? Assume constant angular acceleration while it is starting and stopping.

A car is moving up the side of a circular roller coaster loop of radius 12 m. The angular velocity is 1.8 rad/s and angular acceleration is -0.82 rad/s?. The car is at the same elevation as the center of the loop. Find the magnitude and direction of the acceleration.

An object moves at a constant speed of 9.0 m/s in a circular path of radius of 1.5 m. What is the angular acceleration of the object?

A car accelerates from 0 to 25 m/s in 5 s. If the car’s tires have a diameter of 70 cm, how many revolutions does a tire make during this acceleration?

A point on the edge of a disk rotates around the center of the disk with an initial angular velocity of 3 rad/s clockwise. The graph shows the point’s angular acceleration as a function of time. The positive direction is considered to be counterclockwise. All frictional forces are considered to be negligible.

A discus is held at the end of an arm that starts at rest. The average angular acceleration of [katex]54 \, \text{rad/s}^2 [/katex] lasts for 0.25 s. Find the average tangential acceleration. Find the speed at release. Find the centripetal acceleration just before release. The path is circular and has radius 1.1 m.

In lacrosse, a typical throw is made by rotating the stick through an angle of roughly 90″, then releasing the ball when the stick is vertical, as shown below. If the stick is at rest when horizontal, the length of the stick is 1 meter, and the ball leaves the stick with a velocity of 10 m/s, what angular acceleration must the stick experience? (31.8 rad / 3)

A uniform solid cylinder of mass M and radius R is initially at rest on a frictionless horizontal surface. A massless string is attached to the cylinder and is wrapped around it, as shown in the figure below. The string is then pulled with a constant force F, causing the cylinder to rotate about its center of mass. After the cylinder has rotated through an angle θ, what is the kinetic energy of the cylinder (in terms of F and θ)?

A system consists of two small disks, of masses m and 2m, attached to ends of a rod of negligible mass of length 3x. The rod is free to turn about a vertical axis through point P. The first mass, m, is located x away from point P, and therefore the other mass, of 2m, is 2x from point P. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is . At time t = 0, the rod has an initial counterclockwise angular velocity ωi about P. The system is gradually brought to rest by friction. Derive an expressions for the following quantities in terms of µ, m, x, g, and ωi.

Find the following three values using just rotational kinematics. A ball rolls down a 1.5-meter long incline from rest to 3.2 m/s. The ball has a 16.0 cm radius. Find the angular acceleration of the ball. A CD player spins at 7329 rpm. It starts from rest and has an acceleration of 419 rad/s2. How long does it take to reach full speed? A rotating object has a uniform angular acceleration of 5.15 rad/s2. How long does it take to speed up from 3.33 rad/s to 33.3 rad/s?