# Mass on a Spring

This section contains documents created from scanned original files and other
documents that could not be made accessible to screen reader software. A "#"
symbol is used to denote such documents.

## Video Clips

#### Hooke's Law

RealVideo®
8:49 minutes (1:20 - 10:09)

Statement of the law, with derivation of differential equation for mass on a spring.

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

#### Sinusoidal Motion

RealVideo®
6:42 minutes (10:09 - 16:51)

Sinusoidal motion of mass on a spring proven through demo and differential equation; ω and T determined.

Instructor: Prof. Walter Lewin
Prior Knowledge: Hooke's Law (1:20 of V10)

#### Mass on a Spring Example

RealVideo®
3:38 minutes (16:51 - 20:29)

Calculation of x(t) from initial conditions for mass on a spring.

Instructor: Prof. Walter Lewin
Prior Knowledge: Sinusoidal Motion (10:09 of V10)

#### Hooke's Law Demonstration

RealVideo®
8:38 minutes (20:29 - 29:07)

Period of spring depends on mass but not amplitude; proven by calculation and demo.

Instructor: Prof. Walter Lewin
Prior Knowledge: Sinusoidal Motion (10:09 of V10)

#### Spring and Pendulum Comparison

RealVideo®
4:27 minutes (36:22 - 40:49)

Dependence of T on L, g for pendulum; m, k for spring explained qualitatively.

Instructor: Prof. Walter Lewin
Prior Knowledge: Pendulum Equation (29:07 of V10)

#### Force and Potential Energy

RealVideo®
7:56 minutes (7:00 - 14:56)

Proof that dU/dx=-F using mass on a spring; statement in 3D; application to gravity.

Instructor: Prof. Walter Lewin
Prior Knowledge: Mass on a Spring (1:20 of V10), Gravity (31:11 of V11)

#### Simple Harmonic Motion from Energy Conservation

RealVideo®
5:08 minutes (19:35 - 24:43)

Derivation of differential equation for SHM from conservation of energy in a spring.

Instructor: Prof. Walter Lewin
Prior Knowledge: Force and Potential Energy (7:00 of V13)

#### Energy in a Spring

RealVideo®
5:07 minutes (27:30 - 32:37)

Kinetic and potential energy in spring at equilibrium and at xmax, using conservation.

Instructor: Prof. Walter Lewin
Prior Knowledge: Mass on a Spring (1:20 of V10)

## Lecture Notes

#### Oscillatory Motion

PDF - 1.6 MB#
Page 1 to page 6

Definition and properties of simple harmonic motion; mass-spring systems; energy in simple harmonic motions, with examples.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Hooke's Law, Conservation of Energy, Second Derivatives

#### Hooke's Law

PDF - 1.3 MB#
Page 4 to page 5

Definition of Hooke's law describing restoring force applied by a spring; spring constant of a coil spring; springs in parallel and series.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Newton's Third Law

#### Oscillatory Motion

PDF - 1.6 MB
Page 1 to page 18

Definition and properties of simple harmonic motion; mass-spring systems; energy in simple harmonic motions, with examples; table of equations for simple harmonic motion.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Hooke's Law, Conservation of Energy, Second Derivatives

#### Hooke's Law

PDF#
Page 12 to page 16

Definition of Hooke's law describing restoring force applied by a spring; spring constant of a coil spring; springs in parallel and series.

Instructor: Prof. Stanley Kowalski
Prior Knowledge: Newton's Third Law

#### Mechanical Energy Problem Solving

PDF - 1.0 MB
Page 1 to page 11

Modeling the motion of a block-spring system using Newton's second law and conservation of mechanical energy.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lecture 16

#### Experiment 6

PDF - 1.0 MB
Page 12 to page 25

Harmonic oscillator experiment setup and procedure.

Instructors: Dr. Peter Dourmashkin, Prof. J. David Litster, Prof. David Pritchard, Prof. Bernd Surrow
Prior Knowledge: Lecture 16

## Practice Problems

#### Simple Harmonic Motion

PDF
Problem 1

Momentum and kinetic energy of a baseball bat; simple harmonic motion of two mass-spring systems.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Mass-Spring Oscillation

PDF
Problem 1 to problem 3

Period, acceleration, and amplitude of harmonic motion of mass-spring systems.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Oscillating Mass-Spring

PDF
Problem 3

Velocity of a mass in an oscillating mass-spring system.

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

#### Collisions

PDF
Problem 4 to problem 21

Concept questions about elastic and inelastic collisions between two or more bodies; some questions involve mass-spring systems.

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

#### Hooke's Law

PDF
Problem 2

Stretching of a spring due to hanging masses.

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

#### Experiment 4 Analysis

PDF
Problem 4

Fitting data from Experiment 4.

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

#### Spring-Mass Harmonic Oscillator

PDF
Problem 4

Oscillation of a mass on a spring.

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

#### Inelastic Collision with an Oscillating Mass

PDF
Problem 4

Motion of an oscillating mass on a spring, before and after colliding with a lump of putty.

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

#### Mass on a Spring

PDF#
Problem 5

3-part mass on a spring problem; calculating x for all t, finding v, a, energy for turning point and equilibrium.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Spring with Friction

PDF#
Problem 12

Finding maximum extension, time to maximum velocity for spring extended on frictional surface.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Bungee Jumping

PDF#
Problem 4

Maximum length of cord to protect jumper; distance to water from spring constant.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

#### Distinct Masses Connected by Spring

PDF#
Problem 7

Masses m and 3m are connected by spring; finding energy, velocity, period of oscillations.

Instructor: Prof. Walter Lewin
Prior Knowledge: None

## Exam Questions

#### Mass on a Spring

PDF#
Problem 2

Motion of a mass oscillating on a spring.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Spring-Launched Ball

PDF#
Problem 7

Finding the spring constant of a spring from the maximum height of a ball shot by the spring.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Collision with Mass on a Spring

PDF#
Problem 10

Inelastic collision of a clay ball with a block connected to a spring.

Instructor: Dr. George Stephans
Prior Knowledge: None

#### Spring Compression

PDF#
Problem 12a

Motion of two masses, each connected to a different spring.

Prior Knowledge: None

#### Oscillating Mass on a Spring

PDF#
Problem 14

Harmonic motion of a mass connected to a spring.

Instructor: Dr. George Stephans
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

#### 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

#### Inelastic Collision with a Spring

PDF#
Problem 6

For inelastic collision, finding initial and final velocities and pendulum attributes.

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

#### Mass on a Spring

PDF
Problem 4

Oscillation of a cart connected to a spring on an inclined plane.

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

#### Block and Spring

PDF
Problem 1c

Inelastic collision involving an oscillating mass attached to a spring.

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

#### Spring on an Inclined Plane

PDF#
Problem 1

5-part problem involving Hooke's Law, friction, and conservation of energy.

Instructor: Prof. Walter Lewin
Prior Knowledge: None