Worksheet: Rotational Variables

Q1:

Our solar system orbits the center of the Milky Way Galaxy. Assuming a circular orbit ly in radius and an orbital speed of 250 km/s, how many years does it take for one revolution?

• A years
• B years
• C years
• D years
• E years

Q2:

Each piston in a racing car’s eight-cylinder engine makes a sharp sound every alternate revolution of the engine. The car is driving at a speed where the engine turns through revolutions per kilometer. The repeated sharp sounds produce a sound of frequency 0.750 kHz.

What speed is the racing car driving at?

At how many revolutions per minute is the engine rotating?

• A rpm
• B rpm
• C rpm
• D rpm
• E rpm

Q3:

A boy rides his bicycle which has wheels with a radius of 30.0 cm. If the boy on the bicycle accelerates from rest to a speed of 10.0 m/s in 10.0 s, what is the angular acceleration of the tires?

Q4:

A cyclist is riding such that the wheels of the bicycle have a rotation rate of 3.0 rev/s. If the cyclist brakes such that the rotation rate of the wheels decrease at a rate of 0.3 rev/s2, how long does it take for the cyclist to come to a complete stop?

Q5:

A phonograph turntable rotating at rpm slows down and stops in 1.00 min.

What is the turntable’s angular acceleration assuming it is constant?

How many complete revolutions does the turntable make while stopping?

Q6:

Starting from rest, a wheel spins with a constant angular acceleration of 5.0 rad/s2. The wheel reaches its final angular velocity at the instant that it has turned through a -rad-angular displacement.

What is the final angular velocity of the wheel?

How much time is taken for the wheel to reach its final angular velocity?

Q7:

A circular disk of radius 10 cm has a constant angular acceleration of 1.0 rad/s2. At its angular velocity is 2.0 rad/s.

Determine the disk’s angular velocity at .

What is the angle it has rotated through during this time?

What is the tangential acceleration of a point on the disk at ?

Q8:

A wheel 1.0 m in diameter rotates with an angular acceleration of 4.0 rad/s2, starting with an initial angular speed of 2.0 rad/s and accelerating for 10 seconds in the direction that it is was initially turning.

What is the wheel’s angular speed after accelerating?

Through what angle does the wheel rotate during accelerating?

What is the tangential speed of a point on the rim of the wheel after the wheel has accelerated?

What is the tangential acceleration of a point on the rim of the wheel after the wheel has accelerated?

Q9:

A piece of dust falls onto a spinning compact disk and sticks in place. The spin rate of the disk is rpm and the piece of dust lands 4.30 cm from the disk’s center. What is the total distance traveled by the piece of dust in s of rotation? Assume that the disk is negligibly slowed by the dust falling on it and that the time taken for the dust to accelerate to the same speed as the part of the disk that it lands on is negligible.

Q10:

A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?

Q11:

What is the angular velocity of a 75.0 cm diameter tire on an automobile traveling at 90.0 km/h?

Q12:

The angular velocity of a rotating rigid body increases from 500 rpm to rpm in 120 s.

What is the angular acceleration of the body?

Through what angle does it turn in this 120 s?

Q13:

On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of for 3.0 s.

What is the instantaneous angular speed of the propellers at ?

What is the magnitude of the angular acceleration of the propellers?

Q14:

A rod 2.35 m long is rotating at 1.4 rev/s about an axis at one of its ends.

What is the centripetal acceleration of a point on the rod that is 0.25 m from the rod’s axis of rotation?

What is the centripetal acceleration of a point on the rod that is 1.2 m from the rod’s axis of rotation?

What is the centripetal acceleration of a point on the rod that is 2.0 m from the rod’s axis of rotation?

Q15:

A 5.0-m-diameter wheel is suspended vertically above the ground. While the wheel is at rest, the point at the bottom of the wheel, nearest to the ground, is marked. The wheel then rotates counter clockwise with a constant angular acceleration of 2.0 rad/s2 around a fixed axis through its center.

After 3.0 s of rotation, what is the magnitude of the vertical displacement of the marked point from the base of the wheel?

After 3.0 s of rotation, what is the magnitude of the linear acceleration of the marked point?

Q16:

A compact disc rotates at 350 rpm. The radius of the disc is 150 mm.

What is the tangential speed of a point at the edge of the disc?

What is the tangential speed at a point halfway to the center of the disc?

Q17:

A flywheel slows from 500 rpm to 300 rpm while rotating through 30 revolutions.

What is the magnitude of the angular acceleration of the flywheel?

How much time elapses during the 30 revolutions?

Q18:

A bicycle wheel with radius 0.50 m rotates from rest to an angular speed of 11 rev/s in a time of 7.0 s. What is the magnitude of the total acceleration vector at the edge of the wheel 3.0 s after the wheel starts to accelerate?

Q19:

Calculate the angular velocity of the rotational motion of the Earth around its axis.

Q20:

With the aid of a string, a gyroscope is accelerated uniformly from rest to an angular speed of 56 rad/s, taking 0.50 s.

What is the gyroscope’s angular acceleration?

How many revolutions does gyroscope go through during its acceleration?

Q21:

A wheel rotates at a constant rate of rpm.

What is the wheel’s angular velocity?

Through what angle does the wheel turn in 5.0 s?

Q22:

A flywheel has a radius of 20.00 cm. What is the speed of a point on the edge of the flywheel if it experiences a centripetal acceleration of 9.00 m/s2?

Q23:

A proton in a synchrotron is moving in a circle of radius 825 m. The proton is increasing its speed according to , where m/s and m/s3. What is the magnitude of the proton’s total acceleration at s?

• A m/s2
• B m/s2
• C m/s2
• D m/s2
• E m/s2

Q24:

A flywheel is rotating at 18.7 rev/s. What is the total angle through which a point on the flywheel rotates in 106 s?

Q25:

A particle is executing clockwise circular motion with a constant angular frequency rad/s. The instant corresponds to the particle having an -direction displacement of 2.17 m and a -direction displacement of 0.00 m.

What is the particle’s -direction displacement at the instant ?

What is the particle’s -direction displacement at the instant ?

What is the magnitude of the particle’s centripetal acceleration at the instant ?

Q26:

A rod of length 20 cm has a bead attached to each of its ends. The rod with beads starts rotating from rest and accelerates uniformly. The beads have a tangential speed of 20 m/s after 7.0 s of acceleration. What is the magnitude of the angular acceleration of the rod?

Q27:

A runner is taking part in a 400-m race. Part of the racetrack is curved. When running on the curved portion of the track the runner must run in a circular arc with a radius of curvature of 45.0 m. The runner completes the race in 54.3, moving at constant speed throughout the race. What is the runner’s centripetal acceleration as she runs the curved portion of the track?

Q28:

The tip of a propeller’s blade is 1.5 m from the blade’s axis of rotation. The propeller is at rest at and starts to rotate with a tangential acceleration of the blade tip of 3.00 m/s2. At , what is the total acceleration of the tip of the blade?

Q29:

A boy rides a bicycle a distance of 4.50 km. The bicycle wheels have a radius of 25.0 cm. What is the total angle the tires rotate through during the bike ride?

Q30:

Car A has a speed of 11.3 m/s along a 128-m-radius circular track. Car B has a speed of 9.65 m/s along a 103-m-radius circular track. Calculate the difference in the magnitudes of the centripetal accelerations of Car A and Car B.

Q31:

A particle moves a distance of 3.0 m along a circle of radius 1.5 m in a time of 1.0 s.

Through what angle does the particle rotate around the center of the circle during its motion?

What is the angular speed of the particle?

What is the magnitude of the particle’s angular acceleration?

Q32:

A point located on the second hand of a large clock has a centripetal acceleration of 0.07 cm/s2. How far is the point from the axis of rotation of the second hand?

Q33:

A race car entering the curved part of the track on a race course reduces its speed from 76.2 m/s to 58.6 m/s in 1.73 s. The radius of the curved part of the track is 276 m.

Calculate the magnitude of the total acceleration of the race car at the beginning of its reduction of speed.

Calculate the magnitude of the total acceleration of the race car at the end of its reduction of speed.

Q34:

A particle travels in a circular orbit of radius 13.3 m. The particle’s speed is changing at a rate of 12.6 m/s2 at an instant when its speed is 28.4 m/s. What is the magnitude of the total acceleration of the particle at that instant?

Q35:

A plumb-bob hangs from the roof of a railroad car. The car travels around a circular track of radius 228 m at a speed of 63 km/h. At what angle from the vertical does the plumb bob hang while the car travels around the circular track?

Q36:

A wind turbine is rotating at 0.500 rev/s and slows to a stop in 10.0 s. Its blades are 20.0 m in length.

What is the angular acceleration of the turbine?

What is the centripetal acceleration of the tip of the blades at ?

Q37:

The driver of a car moving at 90.0 km/h presses down on the brake as the car enters a circular curve of radius 150.0 m. If the speed of the car is decreasing at a rate of 9.0 km/h each second, what is the magnitude of the acceleration of the car at the instant its speed is 60.0 km/h?

Q38:

A rotating space station is said to create “artificial gravity”—a loosely defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal acceleration supplied by the floor would allow astronauts to exercise and maintain muscle and bone strength more naturally than in nonrotating space environments. If the space station is 500 m in diameter, what rate of rotation would produce an “artificial gravity” of 9.80 m/s2 at the rim? Give your answer in revolutions per minute.

Q39:

What is the linear speed of Earth as it orbits the Sun, assuming that it follows a circular orbit? Use a value of km for the orbital radius of Earth, and give your answer in kilometers per second.

Q40:

An ultracentrifuge is a device capable of spinning a rotor at very high speeds. Consider an ultracentrifuge rotating at 50 000 rpm. What is the linear speed of a point 0.100 m from its center? Give your answer in kilometers per second.

Q41:

The acceleration due to gravity at Earth’s surface, , is 9.8 m/s2. A satellite orbits Earth at a distance of 350 km above Earth’s surface. What is the acceleration due to gravity at the position of the satellite, as a percentage of ? Use a value of km for the radius of Earth. Give your answer to 3 significant figures.

• A
• B
• C
• D
• E

Q42:

Olympic ice-skaters are able to spin at about 5.00 rev/s.

What is their angular velocity in radians per second?

What is the centripetal acceleration of the skater’s nose if it is 0.120 m from the axis of rotation?

An exceptional skater named Dick Button was able to spin very fast in the 1950s at about 9.00 rev/s. What was the centripetal acceleration of the tip of his nose assuming it is at a 0.120 m radius?

Q43:

Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they experience large centripetal accelerations as they spin, especially at the tip. Consider a helicopter blade that is 4.50 m long and rotates at 350 rpm.

Calculate the magnitude of the centripetal acceleration at the tip of the blade.

What is the ratio of the linear speed of the tip of the blade to the speed of sound, which is taken to be 340 m/s?

Q44:

An ordinary workshop grindstone has a radius of 7.50 cm and rotates at rpm.

Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of g.

What is the linear speed of a point on its edge?

Q45:

The propeller of a World War II fighter plane is 2.50 m in diameter and it spins at rpm.

What is its angular velocity in radians per second?

What is the linear speed of its tip at this angular velocity if the plane is stationary on the tarmac?

What is the centripetal acceleration of the propeller’s tip under these conditions? Calculate it in meters per second squared and convert your answer to multiples of g.

Q46:

The Earth orbits the Sun with an average orbital radius of m. If the Earth’s orbit is modeled as circular and is assumed not to have changed since the Earth was formed, find the total distance that the Earth has moved since its formation. Use a value of years for the time elapsed since the formation of the Earth.

• A m
• B m
• C m
• D m
• E m

Q47:

A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has a 12.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration the magnitude of which is 2.00 times that due to gravity?

Q48:

A truck with 0.420 m radius tires travels at 32.0 m/s.

What is the angular velocity of the rotating tires in radians per second?

What is the rate of rotation of the tires in revolutions per minute?

Q49:

In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 25.0 rad/s and the ball is 1.20 m from the elbow joint, what is the speed of the ball at the instant that it is thrown from the net?

Q50:

A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?

Q51:

Earth has a radius of m at its equator.

What is the period of rotation of Earth in seconds?

What is the rotational angular velocity of Earth? Give your answer to 3 significant figures.

What is the linear velocity of Earth’s surface at the equator?

Q52:

An automobile with 0.260 m radius tires travels km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?

• A rev
• B rev
• C rev
• D rev
• E rev

Q53:

Microwave ovens rotate at a rate of about 9.0 rpm.

What is this rate of rotation in revolutions per second?

What is the angular velocity in radians per second?

Q54:

A semitrailer truck has an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions so it can calculate the distance traveled. If the wheel has a 1.15 m diameter and goes through revolutions, how many kilometers should the odometer read?

Q55:

A race car has 58 cm diameter tires. The car averages a speed of 340 km/h during a race that lasts s. What is the magnitude of the angular displacement of the wheels due to the race?

Q56:

A boy jumps onto a motionless merry-go-round of radius 5.00 m, landing at a point on its rim. The impact of the boy landing on the merry-go-round accelerates it to an angular speed of 5.00 rad/s over a time interval of 20.0 s. What distance does the boy move through over this time?

Q57:

A propellor comes to rest from an angular velocity of 30 rad/s. The change of angular velocity is shown as a function of time in the diagram; the rate of change is constant.

What is the magnitude of the propellor’s angular acceleration?

What angle does the propellor rotate through while coming to rest?

Q58:

A deep-sea fisherman on a boat hooks a big fish on his fishing line. The fish swims away from the boat, pulling the fishing line from the fishing reel, which was initially at rest and has a radius of 4.50 cm. The fishing line unwinds from the reel and, during this process, the reel has a uniform angular acceleration of magnitude 110 rad/s2 for a time of 2.00 s. After this time, the fisherman pulls the fishing line in by winding the reel in the opposite direction to that in which it is pulled by the fish. The magnitude of the angular acceleration produced by the fisherman when he reels in the line is 300 m/s2.

What is the magnitude of the angular velocity of the reel after it is accelerated by the fish pulling on the line?

How many revolutions does the reel make while it is accelerated by the fish pulling on the line?

How much time is required for the fisherman to bring the reel to rest after it is accelerated by the fish pulling on the line?

Q59:

A bicycle mechanic mounts a bicycle on the repair stand and starts the rear wheel spinning from rest. The wheel accelerates uniformly to an angular speed of 250 rpm in a time interval of 5.00 s.

What is the magnitude of the wheel’s angular acceleration?

When the wheel is rotating at 250 rpm, a force is applied to it in a direction that opposes its motion. The force produces an angular acceleration of magnitude 87.3 rad/s2. Over what length time interval must this force be applied to bring the wheel to rest?

Q60:

A flywheel of radius 10.0 cm rotates uniformly through an angle , where .

What is the angular speed of the flywheel?

What angle does the flywheel rotate through in a 30.0 s time interval?

How many revolutions does the flywheel rotate through in a 30.0 s time interval?

What is the tangential speed of a point on the rim of the flywheel during the rotation?

Q61:

A particle moves in a circle of radius 2.00 m. In the time interval 1.50 s < < 4.00 s, the particle’s speed varies with time according to , where m/s and m/s. What is the total acceleration of the particle at the instant s?

Q62:

What is the period of 73.0 Hz electrical power?

Q63:

Find the frequency of a tuning fork that takes s to complete one oscillation.

Q64:

A tire has a tread pattern in which a crevice occurs every 3.50 cm along the tire’s circumference. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at a speed of 41.0 m/s?

• A 1.17 Hz
• B 0.0854 Hz
• C 11.7 Hz
• D 1 170 Hz
• E 1 440 Hz