# Uniform Circular Motion

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## Video Clips

#### Uniform Circular Motion

RealVideo®
6:21 minutes (0:00 - 6:21)

Definition and equations for T, v, ω, and centripetal acceleration, with example.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Loss of Centripetal Force

RealVideo®
6:00 minutes (6:21 - 12:21)

Causes of centripetal acceleration, and demonstration of straight path motion after loss of centripetal force.

Instructor: Prof. Walter Lewin
Prior Knowledge: Circular Motion (beginning of V5)

#### Planets in Circular Orbits

RealVideo®
4:56 minutes (12:21 - 17:17)

Uniform circular motion calculations for planets, and graph showing that ac, and thus gravity, falls off as 1/R2.

Instructor: Prof. Walter Lewin
Prior Knowledge: Centripetal Force (6:21 of V5)

#### Centrifuges

RealVideo®
4:48 minutes (17:17 - 22:05)

Reasoning showing why centrifuges work, with demonstration.

Instructor: Prof. Walter Lewin
Prior Knowledge: Centripetal Force (6:21 of V5)

#### Perceived and Artificial Gravity

RealVideo®
10:06 minutes (22:05 - 32:11)

Perceived gravity defined; examples in several positions including spinning; example of artificial gravity in a spinning space station.

Instructor: Prof. Walter Lewin
Prior Knowledge: Centrifuges (17:17 of V5)

#### Centrifuging a Solution

RealVideo®
9:56 minutes (32:11 - 42:07)

Explanation of how small particles sink to side in a centrifuge; demo of centrifuging AgCl.

Instructor: Prof. Walter Lewin
Prior Knowledge: Centrifuges (17:17 of V5)

#### Water in a Bucket

RealVideo®
8:39 minutes (42:07 - 50:46)

Demonstration of water in a spinning bucket feeling artificial gravity upwards.

Instructor: Prof. Walter Lewin
Prior Knowledge: Centrifuges (17:17 of V5)

#### Newton's First Law

RealVideo®
6:52 minutes (0:00 - 6:52)

Statement of Newton's first law; definition of inertial reference frame; calculation of ac for Earth's rotation.

Instructor: Prof. Walter Lewin
Prior Knowledge: Circular Motion (beginning of V5)

#### Circular Motion

RealVideo®
8:34 minutes (39:39 - 48:13)

Period, frequency, angular velocity, linear speed, centripetal acceleration defined, with example.

Instructor: Prof. Walter Lewin
Prior Knowledge: Circular Motion (beginning of V5)

#### Mechanical Energy and Centripetal Force

RealVideo®
7:34 minutes (23:37 - 31:11)

Example of ball in rollercoaster; proof that h ≥ 2.5R for ball to make it around.

Instructor: Prof. Walter Lewin
Prior Knowledge: Conservation of Energy (17:00 of V11)

## Lecture Notes

#### Uniform Circular Motion

PDF - 1.5 MB#
Page 1 to page 3

Dynamics of uniform circular motion; motion in a vertical circle, with example; conical pendulum, with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Circular Motion, Newton's Laws

#### Angular Momentum

PDF - 1.6 MB#
Page 1 to page 4

Parallel-axis theorem, with example; perpendicular-axis theorem, with examples; definition of angular momentum of a particle; definition of central forces; angular momentum of a particle moving in a straight line; angular momentum of a particle in uniform circular motion; angular momentum of a conical pendulum.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 21

#### Circular Motion

PDF - 1.3 MB#
Page 1 to page 5

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 6

#### Uniform Circular Motion

PDF#
Page 1 to page 6

Dynamics of uniform circular motion; motion in a vertical circle, with example; conical pendulum, with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Circular Motion, Newton's Laws

#### Angular Momentum

PDF#
Page 1 to page 12

Parallel-axis theorem, with example; perpendicular-axis theorem, with examples; definition of angular momentum of a particle; definition of central forces; angular momentum of a particle moving in a straight line; angular momentum of a particle in uniform circular motion; angular momentum of a conical pendulum.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 21

#### Circular Motion

PDF#
Page 1 to page 14

Uniform circular motion; centripetal acceleration; non-uniform circular motion; reference frames; relativity of motion; problem-solving strategy for relative velocities with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 6

#### Projectile and Circular Motion

PDF
Page 1 to page 2

Projectile motion equations; circular motion equations; centripetal acceleration, with equation; imensional analysis method.

Instructor: Dr. George Stephans
Prior Knowledge: Multi-Dimensional Kinematics

#### Circular Motion

PDF
Page 1 to page 10

Position and displacement of object in circular orbit; magnitude of displacement; magnitude of velocity and angular velocity; direction of velocity; tangential acceleration defined; uniform circular motion; period and frequency of orbit.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Newton's Laws

#### Circular Motion

PDF
Page 1 to page 12

Summary of circular motion, with equations; circular motion vector description, with equations; circular motion modeling problems; analysis of acceleration in circular motion.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lecture 9

#### Circular Dynamics Review

PDF
Page 1 to page 11

Position vector of circular orbit; angular velocity and tangential velocity; radial acceleration; uniform circular motion; non-uniform circular motion; comparison of linear and circular motion at constant acceleration; problem solving strategy involving circular motion.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lecture 9

#### Rotational Kinematics Review

PDF
Page 9 to page 12

Fixed axis rotational kinematics, with definitions of angular velocity, angular acceleration, tangential velocity, tangential acceleration, and radial acceleration.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lectures 22, 24

## Practice Problems

#### Model Airplane on a Wire

PDF
Problem 6

3-part problem; calculating tension in wire, critical angle, and safe speed for flight.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

PDF
Problem 4

3-part problem; finding normal force from wire and height the bead attains on the way back up.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Uniform Circular Motion

PDF
Problem 1

Speed and acceleration of an object moving in a circle of constant radius.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Circular Motion with Gravity

PDF
Problem 1

Forces and acceleration of objects moving in vertical circular paths.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Angular Speed

PDF
Problem 1

Relative angular speed of two objects at different points on a merry-go-round.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Whirling Objects

PDF
Problem 11

Forces acting on two bodies rotating around a shaft with constant angular velocity.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion Inside a Cone

PDF
Problem 12

Motion of an object in a circular orbit along the inside of a cone.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Car on a Banked Turn

PDF
Problem 13

Finding velocity needed to keep a car from slipping on a banked turn.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### U-Control Model Airplane

PDF
Problem 14

Motion of a plane tethered to a fixed point.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion and Conservation of Mechanical Energy

PDF
Problem 23

Motion of an object along a frictionless loop-the-loop track.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Beads Sliding Along a Ring

PDF
Problem 23

Forces acting on a suspended ring with sliding beads.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion

PDF
Problem 1

Angular velocity of two bugs on a merry-go-round.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion

PDF
Problem 2

Linear acceleration of a bug on a merry-go-round.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion

PDF
Problem 3

Tangential acceleration of a bug on a merry-go-round.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion and Reference Frames

PDF
Problem 4

Motion of a ball thrown from a merry-go-round.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Force and Friction

PDF
Problem 5

Minimum angular speed required for "barrel of fun."

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Non-Uniform Circular Motion

PDF
Problem 6

Speed of an object moving in a circular path with non-uniform acceleration.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Accelerating Circular Motion

PDF
Problem 7

Direction of acceleration vector for object in circular motion with increasing speed.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Force

PDF
Problem 8

Forces acting on a passenger in a car navigating a circular turn.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Acceleration

PDF
Problem 9

Forces capable of producing centripetal acceleration.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Force

PDF
Problem 10

Source of centripetal force for a ball moving on turning truck bed.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Car Wreck

PDF
Problem 11

Determining spinning motion of a wrecked car.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Circular Motion

PDF
Problem 13

Acceleration of ice sliding in a bowl.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Uniform Circular Motion

PDF
Problem 14

Tension of a string holding a stone moving in a vertical circular path.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Acceleration

PDF
Problem 15

Acceleration of a puck moving in a uniform circle on a table.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Centripetal Acceleration

PDF
Problem 16

Acceleration of a puck moving in a uniform circle on a table with a shorter lever arm.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Vertical Circular Motion

PDF
Problem 17

Tension in a string holding a ball being swung in a vertical circle.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Bicycle Tire

PDF
Problem 18

Speed of a point on the outside of a tire in relation to the speed of the center of mass.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Bicycle Tire

PDF
Problem 19

Relative speeds of points on the outside of a rolling tire.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Bicycle Tire

PDF
Problem 20

Direction of motion of points on the outside of a rolling tire.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Turntable

PDF
Problem 21

Forces acting on a cube lying on a spinning turntable.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Spinning Ball

PDF
Problem 4

Tension in strings suspending a ball from a spinning vertical rod.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Rolling and Work

PDF
Problem 11

Work done on a ball rolling in a spiral path along the inside of a hollow cylinder.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Experiment 4 Pre-Lab

PDF
Problem 5

Finding the spring constant of a spring; finding the radius of an object in uniform circular motion.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Rotational Kinematics

PDF
Problem 1

Velocity and centripetal acceleration of a person on the Earth's surface.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Banked Turn

PDF
Problem 2

Motion of a car traversing a banked circular turn.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Tetherball

PDF
Problem 3

Motion of a ball suspended from a string and travelling in a vertical circle.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: None

#### Planetary Rotation

PDF#
Problem 5

Finding maximum rotation period for planet from density; calculating periods.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

## Exam Questions

#### Circular Racetrack

PDF#
Problem 2

Maximum speed before skidding for car on frictional track for two different masses; Angular velocity.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Uniform Circular Motion

PDF#
Problem 3

Period of a rotating disc attached to a spring.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Circular Motion of a Car

PDF#
Problem 7

Motion of a car driving along the inside of a hollow cone.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Spinning Rod and Mass

PDF#
Problem 10

Dynamics of a system of a mass suspended from a vertical spinning rod by a string and pulley.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Equilibrium of a Runner

PDF
Problem 4

Forces and torques acting on a runner rounding a circular track.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Circular Motion in a Cone

PDF#
Problem 2

4-part problem; drawing free-body diagram and finding v, T for object; maximum speed with friction.

Instructors: Dr. Peter Dourmashkin, Prof. Kate Scholberg
Prior Knowledge: None

#### Experimental Centripetal Force

PDF#
Problem 4A

Finding ω, T, f for nut revolving around axis on end of a rubber band.

Instructors: Dr. Peter Dourmashkin, Prof. Kate Scholberg
Prior Knowledge: None

#### Ball in a Vertical Circle

PDF#
Problem 2

Speed and tension at the top of the circle for ball swinging around.

Instructors: Dr. Peter Dourmashkin, Prof. Kate Scholberg
Prior Knowledge: None

#### Marooned on an Asteroid

PDF#
Problem 3

4-part problem; finding compression necessary to launch pen into orbit; speed and radius in orbit.

Instructors: Dr. Peter Dourmashkin, Prof. Kate Scholberg
Prior Knowledge: None

#### Water Bucket

PDF
Problem 2

Motion of a water bucket spun in a vertical circle.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Uniform Circular Motion

PDF
Problem 1a

Centripetal acceleration of a puck moving in a circle.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Circular Motion

PDF
Problem 1b

Speed of an object with a constantly changing velocity vector.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None

#### Conservation of Energy

PDF
Problem 5

Kinematics and dynamics of a tethered ball traveling in a vertical circle.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: None