# Torque & Rotational Statics

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## Video Clips

#### Conservation of Angular Momentum

RealVideo®
9:18 minutes (8:18 - 17:36)

Prove that dL/dt = r × F; angular momentum conserved when τ = 0; Spin angular momentum defined.

Instructor: Prof. Walter Lewin
Prior Knowledge: Angular Momentum (beginning of V20)

RealVideo®
8:26 minutes (0:57 - 9:23)

Critical angle calculated for ladder resting against a wall, based on μ.

Instructor: Prof. Walter Lewin
Prior Knowledge: F=ma (6:52 of V6) and Torque

RealVideo®
10:24 minutes (9:23 - 19:47)

Ladder stability as person walks from base to center of mass and further; overcoming maximal friction.

Instructor: Prof. Walter Lewin
Prior Knowledge: Ladder Stability I (0:57 of V25)

RealVideo®
2:03 minutes (19:03 - 21:06)

Demonstration of weight placed on ladder near critical angle at various positions.

Instructor: Prof. Walter Lewin
Prior Knowledge: Ladder Stability II (9:23 of V25)

#### Equilibrium and Center of Mass

RealVideo®
9:36 minutes (33:54 - 43:30)

Statics of a hanging object; position of center of mass for equilibrium; stability of several small objects.

Instructor: Prof. Walter Lewin
Prior Knowledge: Static Equilibrium (beginning of V6)

#### Rope Walker

RealVideo®
4:35 minutes (43:30 - 48:05)

Stability of rope walker with objects suspended to lower center of mass; demonstration of rope walker on zip line.

Instructor: Prof. Walter Lewin
Prior Knowledge: Equilibrium and Center of Mass (33:54 of V25)

#### Atwood's Machine

RealVideo®
7:51 minutes (7:43 - 15:34)

Calculation of acceleration for two blocks suspended over pulley with friction from masses and moment of inertia.

Instructor: Prof. Walter Lewin
Prior Knowledge: Torque, F=ma (6:52 of V6)

## Lecture Notes

#### Rotational and Translational Motion

PDF#
Page 1

Summary of analogies between rotation and linear motion.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Conservation of Angular Momentum

PDF - 1.6 MB#
Page 4 to page 6

Relationship between forces and angular momentum; definition of the law of conservation of angular momentum; definition of torque; angular momentum and central forces, with examples.

Prior Knowledge: Angular Momentum, cross products

#### Rigid Body Dynamics

PDF - 1.7 MB#
Page 1 to page 6

Torque and force; angular momentum and torque; torque and angular momentum of a conical pendulum; torque and angular acceleration; rigid body angular acceleration, with examples; torque due to gravity, with examples; conservation of angular momentum.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 22

#### Rigid Body Dynamics Examples

PDF - 1.6 MB#
Page 1

Rotational dynamics examples, including particle on a string and spinning bicycle wheel.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 23

#### Work and Energy in Rotational Motion

PDF - 1.6 MB#
Page 1 to page 6

Work-energy theorem in rotational motion, with examples; angular impulse, with example; decomposition of displacement into translation and rotation; rolling motion of cylinders and spheres.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 23

#### Rolling of Rigid Bodies

PDF - 1.3 MB#
Page 1 to page 5

Rolling motion of a rigid body, such as a cylinder or a sphere, with several examples; angular momentum and collisions, with example; gyroscope precession.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 24

#### Forces in Equilibrium

PDF - 1.6 MB#
Page 1 to page 6

Conditions of equilibrium for particles and rigid bodies; torque axis; center of gravity; center of mass demonstration; stable and unstable equilibrium; coupled forces; equilibrium problem-solving strategy with several examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 25

#### Conservation of Angular Momentum

PDF
Page 13 to page 18

Relationship between forces and angular momentum; definition of the law of conservation of angular momentum; definition of torque; angular momentum and central forces, with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Angular Momentum, Cross Products

#### Rigid Body Dynamics

PDF
Page 1 to page 20

Torque and force; angular momentum and torque; torque and angular momentum of a conical pendulum; torque and angular acceleration; rigid body angular acceleration, with examples; torque due to gravity, with examples; conservation of angular momentum.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 22

#### Rigid Body Dynamics Examples

PDF
Page 1 to page 3

Rotational dynamics examples, including particle on a string and spinning bicycle wheel.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 23

#### Work and Energy in Rotational Motion

PDF
Page 4 to page 19

Work-energy theorem in rotational motion, with examples; angular impulse, with example; decomposition of displacement into translation and rotation; rolling motion of cylinders and spheres.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 23

#### Rolling of Rigid Bodies

PDF
Page 1 to page 17

Rolling motion of a rigid body, such as a cylinder or a sphere, with several examples; angular momentum and collisions, with example; gyroscope precession.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 24

#### Forces in Equilibrium

PDF - 1.5 MB
Page 1 to page 15

Conditions of equilibrium for particles and rigid bodies; torque axis; center of gravity; center of mass demonstration; stable and unstable equilibrium; coupled forces; equilibrium problem-solving strategy with several examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Lecture 25

#### Torque

PDF
Page 1 to page 2

Torque defined; conditions for rotational static equilibrium; torque and rotational dynamics; moment of inertia; parallel axis theorem.

Instructor: Dr. George Stephans
Prior Knowledge: Rotational Kinematics

#### Angular Momentum

PDF
Page 1 to page 2

Checklist for solving torque problems; angular momentum defined, with equation; angular impulse defined, with equation.

Instructor: Dr. George Stephans
Prior Knowledge: Rotational Dynamics, Rotational Energy

#### Torque

PDF#
Page 9 to page 25

Rigid bodies; center of mass; pivoted lever; lever law; pivoted lever example problem; generalized lever law; definition of torque; torque sign convention; two geometric interpretations of torque; static equilibrium.

Prior Knowledge: Cross Products, Newton's Laws

#### Statics and Torque Review

PDF
Page 12 to page 18

Definition of torque, including magnitude, direction, and sign convention; moment arm of a force; torque from tangential force; static equilibrium.

Prior Knowledge: Lecture 12

## Practice Problems

#### Rotational Statics

PDF#
Problem 1

Static equilibrium involving torques.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Cross Product

PDF#
Problem 1

Finding the cross product of two vectors.

Prior Knowledge: None

#### Wrench and Nut

PDF#
Problem 2

Effectiveness of various methods of applying a force to a wrench to loosen a nut.

Prior Knowledge: None

PDF#
Problem 3

Forces acting on a uniform ladder leaning against a wall.

Prior Knowledge: None

#### Opening a Door

PDF#
Problem 4

Torque applied to a stuck door by hand and by using a rope.

Prior Knowledge: None

#### Balanced Rock

PDF#
Problem 5

Finding the weight of a measuring stick in order to balance a suspended rock.

Prior Knowledge: None

#### Center-of-Mass

PDF#
Problem 6

Position of the center-of-mass of a box on an inclined plane as it relates to the box tipping over.

Prior Knowledge: None

#### Equilibrium in an Ankle

PDF#
Problem 7

Torque acting on an ankle.

Prior Knowledge: None

#### Equilibrium in a Hip

PDF#
Problem 8

Torque acting on a hip joint.

Prior Knowledge: None

#### Pulling a Cylinder onto the Sidewalk

PDF#
Problem 1

2-part problem; calculating force required and optimal angle for applying force.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

## Exam Questions

#### Rigid Rotor

PDF#
Problem 2

4-part problem; finding center of mass, moment of inertia of rotor; α from gravity and ω at vertical.

Instructor: Prof. Stanley Kowalski
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

#### Equilibrium of a Pushup

PDF
Problem 5

Forces and torques acting on a person doing a pushup.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Static Equilibrium

PDF
Problem 6

Forces and torques acting on a plank holding a mass of bricks and supported by cables.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Static Equilibrium

PDF
Problem 7

Forces and torques acting on a plank leaning against a wall.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Torque and Angular Acceleration

PDF#
Problem B7

Two masses connected by a rope wrapped around a pulley.

Prior Knowledge: None