Breadcrumbs

About

Intro Page

Newton's cradle

Newton's cradle is a device that demonstrates conservation of momentum and energy using a series of swinging spheres. When one sphere at the end is lifted and released, it strikes the stationary spheres, transmitting a force through the stationary spheres that pushes the last sphere upward. The last sphere swings back and strikes the still nearly stationary spheres, repeating the effect in the opposite direction. The device is named after 17th-century English scientist Sir Isaac Newton. It is also known as Newton's pendulum, Newton's balls, Newton's rocker or executive ball clicker (since the device makes a click each time the balls collide, which they do repeatedly in a steady rhythm).

A typical Newton's cradle consists of a series of identically sized metal balls suspended in a metal frame so that they are just touching each other at rest. Each ball is attached to the frame by two wires of equal length angled away from each other. This restricts the pendulums' movements to the same plane.

Controls

The first comboBox allows eay selection of number of balls in the Newton's cradle

1 ball to the left

2 balls to the left

3 balls to the left

4 balls to the left

5 balls to the left

6 balls to the left

7 balls to the left

1 ball 1 raises 1 ball to the left and 1 ball to the right

2 ball 1 raises 2 ball to the left and 1 ball to the right

2 ball 2 raises 2 balls to the left and 2 balls to the right

3 ball 3 raises 3 balls to the left and 3 balls to the right

n is the number of balls allowed

i is the index of the ball to change

for which m[i] is the mass of the index ball

play button starts the simulation

reset button starts the simulation at afresh

Physics explanation

Newton's cradle can be modeled fairly accurately with simple mathematical equations with the assumption that the balls always collide in pairs.

The motion of the balls are modelled using dθ[i]dt=ω[i]

where θ is the angle of the ball

ω is the angular velocity

i represents each element of the ball

t is time

the second order differential equation is d(ω[i])dt=−g∗sin(θ[i])L−k∗L∗ω[i]

where g is gravitational constant 9.81 m/s2

L is the length of the inextensible string connecting the ball to the support for rotational motion

k is a resistant coefficient to rotational motion ω

Events of the ordinary differential equations

In a state event, a TOLERANCE =1.0e-6 is defined, a loop to check through each ball i, where a check of the θ[i] is greater than θ[i+1] plus the TOLERANCE, if true, return the balls to their closed up angular position θ[i+1]−θ[i]. If false, do nothing.

Fixed Relationship Calculations

A for loop over i=0 to i less than number of balls to calculate the following

dx[i]=L∗(sinθ[i]) where dx[i] is the change is x[i] position

dy[i]=−L∗cos(θ[i]) where dy[i] is the change in y[i] position

x[i]=xc[i]+dx[i] where xc[i] is each balls centre or equilibrium x position

y[i]=ymax+dy[i] where ymax is the support y position

vx[i]=L∗ω[i]∗cos(θ[i]) where vx[i] is the x direction linear velocities of each ball

vy[i]=L∗ω[i]∗sin(θ[i]) where vy[i] is the y direction linear velocities of each ball

Translations

Code	Language		Translator	Run

Credits

Fu-Kwun Hwang and lookang (This email address is being protected from spambots. You need JavaScript enabled to view it.); lookang; Francisco Esquembre

Check out Glowing Newton's Cradle Desk Ornament Small And Large Sizes With Led Lights. Get it on Shopee now! https://shope.ee/3AhkYwOP03?share_channel_code=1

20231223 Update on WEBEJS does not work due to EJS5.33, need some time to fix convert to EJS6.02: https://macmath.inf.um.es/editor?id=lookang&url=https%3A//iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejss_model_newtonscadlerealwee3d.zip

What is a Newton's cradle and what principles does it demonstrate?

A Newton's cradle is a device consisting of a series of suspended metal balls. When one ball is lifted and released, it strikes the stationary balls, transferring momentum and energy through them to the last ball, causing it to swing upward. This process repeats in a rhythmic back-and-forth motion. It demonstrates the conservation of momentum and energy, specifically focusing on how these quantities are transferred in collisions between objects.

How does the simulation model the motion of the balls in the Newton's cradle?

The simulation models the motion of the balls using differential equations that describe the angular position (θ) and angular velocity (ω) of each ball. These equations consider factors like gravity (g), the length of the strings (L), and air resistance (k). The simulation also includes a state event that checks for collisions by comparing the angles of adjacent balls, triggering an action that simulates a perfectly elastic collision and conserves angular momentum. In short, the balls move according to Newtonian mechanics (F=ma) and their interaction is modelled as a series of elastic collisions.

What controls are available in the Newton's cradle simulation?

The simulation provides controls for selecting the number of balls in the cradle, including options to raise multiple balls simultaneously. Users can also select a specific ball to lift and release, explore varying the mass of each individual ball, and start/reset the simulation, and observe the simulation in both 2D and 3D. It is possible to "grab" and move any of the balls in the system.

What mathematical equations are used to model the physics of the Newton's cradle?

The simulation uses a system of differential equations to describe the motion of the balls. Key equations include:

dθ[i]/dt = ω[i] describes how the angular position changes with time and angular velocity.
d(ω[i])/dt = -g*sin(θ[i])/L - k*L*ω[i] describes the angular acceleration and accounts for gravitational force and air resistance. The simulation also considers how the positions and velocities of the balls are updated, the transfer of momentum and kinetic energy during collisions, and models for position in both the X and Y axis.

How are collisions handled in the simulation?

Collisions are handled as state events which are triggered when the angular positions of two adjacent balls are determined to be less than a predefined tolerance. A perfectly elastic collision model is used to adjust the angular velocities of the colliding balls, conserving angular momentum. Specifically the simulation uses the following equations: va=((m1-m2)v1+2m2v2)/(m1+m2); vb=(2m1*v1+(m2-m1)*v2)/(m1+m2); where v1 and v2 are the angular velocities before and va and vb are the angular velocities after the collision of the two balls with masses m1 and m2.

What is meant by the "fixed relationship calculations" in the simulation's physics explanation?

The "fixed relationship calculations" refer to how the linear positions and velocities of the balls are derived from their angular positions and velocities. These calculations ensure that the balls move along the correct paths as pendulums by using trigonometric relationships for deriving linear displacement and velocity components (dx, dy, vx, vy) based on angular displacement and angular velocity.

What exercises are suggested for users of the Newton's cradle simulation?

The simulation encourages users to conduct several experiments. These include:

Observing the behavior of the cradle with 2, 3, 4, 5, 6, and 7 balls and explaining these results mathematically in terms of conservation of momentum and energy.
Comparing the virtual model with a real world apparatus, identifying differences due to air resistance, friction and other sources of loss.
Exploring the effect of varying ball masses to see how changing masses impacts the results. The intention is to prompt users to evaluate how well the simulation models the laws of physics it is intended to represent.

How can advanced learners further engage with the Newton's cradle model?

Advanced learners are encouraged to "remix" the model by adding new features not included in the original simulation and sharing their updated simulation through the NTNUJAVA Virtual Physics Laboratory. Examples of these features might include adding ways to simulate or visualise the impact of air resistance or varying other parameters to explore the limitations of the model and assumptions.

Newton's Cradle Study Guide

Quiz

Instructions: Answer the following questions in 2-3 sentences each.

What is the main principle that Newton's Cradle demonstrates?
Describe the typical physical arrangement of a Newton's Cradle.
What is the purpose of the angled wires that suspend the balls?
Explain how the number of balls affects the outcome of the simulation, based on the provided information.
In the context of the simulation, what do the variables θ and ω represent?
What is the significance of the gravitational constant 'g' in the model's equations?
Explain the purpose of the 'state event' in the simulation's code.
What does the simulation use to model collisions between the balls?
How are the linear velocities of the balls (vx[i], vy[i]) calculated in the simulation?
What are some real-world factors that are not taken into account in this simulation?

Quiz Answer Key

Newton's Cradle primarily demonstrates the conservation of momentum and energy. When one ball strikes the rest, momentum and kinetic energy are transferred through the series, resulting in the end ball swinging outward.
A typical Newton's Cradle consists of several identically sized metal balls suspended in a frame so they are just touching when at rest. Each ball is hung by two wires of equal length, angled away from each other.
The angled wires restrict the movement of the pendulums to a single plane. This ensures that the balls collide in a straight line, simplifying the demonstration of momentum and energy transfer.
The simulation allows for variation in the number of balls that are set in motion. It can simulate scenarios with one to seven balls in motion on one side or, it can set in motion a different number of balls on each side of the cradle at the same time.
In the simulation, θ represents the angle of each ball relative to its resting position, and ω is the angular velocity, which is the rate of change of the angle. These variables are crucial in describing the motion of each pendulum.
The gravitational constant 'g' is used in the simulation to account for the force of gravity, which acts to pull the balls back to their resting position. This is a fundamental component in the pendulum-like movement of each ball.
The 'state event' code checks if the angle of one ball (θ[i]) is greater than the angle of the ball to its right (θ[i+1]) plus a tolerance value. If true, it returns the balls to their closed up angular position, modeling the collision event.
The simulation models collisions as perfectly elastic in one dimension. It uses the principles of conservation of momentum and angular momentum to determine the velocities of the balls after they collide.
The linear velocities of the balls are calculated based on the angular velocity (ω[i]), the length of the pendulum string (L), and the angle of the pendulum (θ[i]). Vx[i] = L * ω[i] * cos(θ[i]) and vy[i] = L * ω[i] * sin(θ[i])
This simulation does not account for real-world factors such as air resistance, friction at the point of suspension, sound production and the minute amount of vibrations from collisions. These factors would gradually reduce the motion of the real-world device.

Essay Questions

Instructions: Answer each question in a well-organized essay format.

Discuss how the Newton's Cradle effectively demonstrates the laws of conservation of momentum and energy. Include specific references to the equations and principles outlined in the source material.
Analyze the assumptions made in the mathematical model of the Newton's Cradle presented in the document. How do these assumptions simplify the real-world behavior of the apparatus, and what are the implications of these simplifications?
Compare and contrast the way that the simulation's model handles collision events with how collisions would happen in a real-world scenario. Be sure to address the concepts of ideal versus non-ideal systems in your answer.
How does the simulation's controls and design features allow for users to explore the physics of the Newton's Cradle? Consider the adjustable variables, views, and exercises presented in the document.
Beyond conservation of momentum and energy, what other areas of physics could be studied using a Newton’s Cradle model? Consider areas such as oscillation, wave mechanics, and others.

Glossary of Key Terms

Conservation of Momentum: A fundamental principle in physics stating that the total momentum of a closed system remains constant if no external forces act upon it.
Conservation of Energy: A principle stating that the total energy of an isolated system remains constant; energy can neither be created nor destroyed but can transform from one form to another.
Momentum: A property of a moving object, defined as the product of its mass and velocity (p=mv).
Kinetic Energy: The energy that a body possesses due to its motion (KE = 1/2mv²).
Angular Velocity (ω): The rate of change of the angular position of a rotating body, measured in radians per second.
State Event: In a simulation, a state event occurs when a particular condition is met (e.g., two balls colliding), triggering a change or action in the system.
Ordinary Differential Equation (ODE): An equation involving a function of one independent variable and its derivatives, used to model continuous change over time.
Gravitational Constant (g): The physical constant representing the strength of the gravitational force, approximately 9.81 m/s^2 on Earth.
Resistance Coefficient (k): A measure of a body's resistance to movement, used in the simulation to account for effects such as air resistance.
TOLERANCE: In this context, a numerical threshold (1.0e-6) used to determine when a collision is considered to occur in the simulation by comparing the angles of adjacent balls.
Elastic Collision: A type of collision where both momentum and kinetic energy are conserved.
Inextensible String: A string that does not stretch; it maintains a constant length.

Description

Newton's Cradle Model

reference: http://en.wikipedia.org/wiki/Newton%27s_cradle
Newton's cradle, named after Sir Isaac Newton, is a device that demonstrates conservation of momentum and energy via a series of swinging spheres.

Construction in 3D
A typical Newton's cradle consists of a series of identically sized metal balls suspended in a metal frame so that they are just NOT touching each other at rest. Each ball is attached to the frame by two wires of equal length angled away from each other. This restricts the pendulums' movements to the same plane.

Mathematical Model:

the simplified equations of motion is ODE for a single mass(pendulum) are:
d (cta)/dt = omega
d(omega)/dt = -g*sin(cta)/L

when extended to n array of masses, for for(int i=0;i<n;i++){
d (cta[i])/dt = omega[i]
d(omega[i])/dt = -g*sin(cta[i])/L // simplified add here from air drag
}
the zero condition type as "State event" with the following code

// start of code for zero condition
double min = TOLERANCE;
for(int i=0;i<n-1;i++){
if(cta[i]>cta[i+1]+TOLERANCE){//<TOLERANCE){// collision between i and i+1;
cid=i;
return cta[i+1]-cta[i];
}
}
return TOLERANCE;// no collision
//end of code for zero condition

The action is to assumes the collision are modeled by perfectly elastic collision in one dimension with angular momentum conserved
the code is as below.
// start of code for action
m1=m[cid];
m2=m[cid+1];
v1=L*omega[cid];
v2=L*omega[cid+1];
va=((m1-m2)*v1+2*m2*v2)/(m1+m2);// velocity after collision
vb=(2*m1*v1+(m2-m1)*v2)/(m1+m2);
omega[cid]=va/L;// back to omega
omega[cid+1]=vb/L;
//end of code for action

Design features:

currently, the simulation allows of exploration of n max = 7 masses with each mass m[i] that can be vary from the slider control from 1 to 5 kg.
additional variables are:
gravitational constant downward g = 9.81 m/s^2
Length of pendulums, L
k coefficicent of air resistance model by equation F = k*omega

The usabilty control:
To move the balls, move the cursor over to the mass, click and hold the left mouse and drag the mouse to lift the masses. note that all masses are selectable and movable to a new height along the each path of the pendulums swing.

Design Views:

there are 2 types views available selectable by the check-boxes 2D and 3D,
2D world view
3D world view
hint:
may be useful to observe the transfer of linear momentum from ball to ball by pressing the step button to aid understanding of momentum transfer during collisions
may be useful to observe the scientific representations of kinetic energies KE[i], gravitational potential energies PE[i] and total mechanical energies TE[i]
graph of scientific representations of kinetic energies KE total, gravitational potential energies PE total and total mechanical energies TE total

Exercises:

select n =2 for a simple 2 ball system for the experiment.

Pull and release one ball. Note the results and explain mathematically in terms of conservation of engergy and momentum.
m1 u1 + m1 u1 = m1 v1+ m2 v2
1/2 m1 u1^2 + 1/2 m2 u2^2 = 1/2 m1 v1^2 + 1/2 m2 v2^2
record your observations and discuss if the two equations above can account for the observations

select n = 3 , 4, 5, 6, 7 and repeat the experiment above.
Do the results meet your expectations?
hint: can you conclude that the just before and just after collisions
momentum in = momentum out
kinetic energies in = kinetic energies out.
the clue lies in the conservation of momentum and kinetic energies just before and after collisions.

Discuss what are the differences between this computer model and real life apparatus.
hint: the newton's cradle motion will continue in this back and fro motion until all energies are lost to damping due to air resistance, friction, sound and vibrations.

In this simulation the default is all balls have the same mass. What would occur if this was not the case?

Advanced Learner:
Please submit your remix model that model features that are not available in the existing virtual lab and share your model with the world through NTNUJAVA Virtual Physics Laboratory http://www.phy.ntnu.edu.tw/ntnujava/index.php?board=28.0. Impacting the world with your model now.

Credits:

The Newton's Cradle Model was created by Fu-Kwun Hwang, customized by Loo Kang WEE, using the Easy Java Simulations (EJS) version 4.3.3.2 authoring and modeling tool. An applet version of this model is available on the NTNU website < http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=2195.0 >.

You can examine and modify this compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open EJS Model" from the pop-up menu. You must, of course, have EJS installed on your computer. Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.

Version:

http://weelookang.blogspot.sg/2015/06/ejss-newtons-cradle-model.html by lookang

Other Resources

http://www.walter-fendt.de/html5/phen/newtoncradle_en.htm by Walter Fendt

FAQ

What is a Newton's cradle and what principles does it demonstrate?

How does the simulation model the motion of the balls in the Newton's cradle?

What controls are available in the Newton's cradle simulation?

What mathematical equations are used to model the physics of the Newton's cradle?

The simulation uses a system of differential equations to describe the motion of the balls. Key equations include:

dθ[i]/dt = ω[i] describes how the angular position changes with time and angular velocity.
d(ω[i])/dt = -g*sin(θ[i])/L - k*L*ω[i] describes the angular acceleration and accounts for gravitational force and air resistance. The simulation also considers how the positions and velocities of the balls are updated, the transfer of momentum and kinetic energy during collisions, and models for position in both the X and Y axis.

How are collisions handled in the simulation?

What is meant by the "fixed relationship calculations" in the simulation's physics explanation?

What exercises are suggested for users of the Newton's cradle simulation?

The simulation encourages users to conduct several experiments. These include:

Observing the behavior of the cradle with 2, 3, 4, 5, 6, and 7 balls and explaining these results mathematically in terms of conservation of momentum and energy.
Comparing the virtual model with a real world apparatus, identifying differences due to air resistance, friction and other sources of loss.
Exploring the effect of varying ball masses to see how changing masses impacts the results. The intention is to prompt users to evaluate how well the simulation models the laws of physics it is intended to represent.

How can advanced learners further engage with the Newton's cradle model?

3.5 1 1 1 1 1 1 1 1 1 1 Rating 3.50 (3 Votes)

Details: Parent Category: 03 Motion & Forces; Category: 02 Dynamics; Published: 06 June 2016; Created: 20 August 2015; Last Updated: 18 December 2024; Hits: 15163

Breadcrumbs

📔Newton Cradle (3D) JavaScript HTML5 Applet Simulation Model

About

Newton's cradle

Controls

Physics explanation

Events of the ordinary differential equations

Fixed Relationship Calculations

Translations

Credits

What is a Newton's cradle and what principles does it demonstrate?

How does the simulation model the motion of the balls in the Newton's cradle?

What controls are available in the Newton's cradle simulation?

What mathematical equations are used to model the physics of the Newton's cradle?

How are collisions handled in the simulation?

What is meant by the "fixed relationship calculations" in the simulation's physics explanation?

What exercises are suggested for users of the Newton's cradle simulation?

How can advanced learners further engage with the Newton's cradle model?

Newton's Cradle Study Guide

Quiz

Quiz Answer Key

Essay Questions

Glossary of Key Terms

Description

Newton's Cradle Model

Mathematical Model:

Design features:

Design Views:

Exercises:

Credits:

Version:

Other Resources

FAQ

What is a Newton's cradle and what principles does it demonstrate?

How does the simulation model the motion of the balls in the Newton's cradle?

What controls are available in the Newton's cradle simulation?

What mathematical equations are used to model the physics of the Newton's cradle?

How are collisions handled in the simulation?

What is meant by the "fixed relationship calculations" in the simulation's physics explanation?

What exercises are suggested for users of the Newton's cradle simulation?

How can advanced learners further engage with the Newton's cradle model?