Resource Lesson
Systems of Bodies
Printer Friendly Version
When two or more bodies are connected by a cord and move in tandem, then they are considered to be a
system of bodies
. When working problems involving systems of bodies,
the first step is to draw
freebody diagrams
for each object. Remember, that the principle forces that we are now considering are: normals, weight, friction, tensions and generic applied forces.
since both masses are attached, they share the same kinematics properties: displacement, velocity, acceleration and time.
when writing equations for net F = ma, the direction of motion is considered to be the positive direction for the acceleration of the objects comprising the system.
Table Surface
Let's consider as our first example, two identical objects being dragged across the surface of a frictionless table by two cords.
The two freebody diagrams would look like:
left mass
right mass
In order to determine the acceleration of the system and the tension in each cord, we will need to write the equations of motion for each mass: net F
_{x}
= ma
_{x}
and net F
_{y}
= ma
_{y}
.
left mass
right mass
net F
_{x}
= ma
_{x}
T
_{1}
= ma
T
_{2}
- T
_{1}
= ma
net F
_{y}
= ma
_{y}
- mg = 0
- mg = 0
Solving the two equations for net F
_{x}
= ma
_{x}
simultaneously yields the equation:
T
_{1}
= ma
T
_{2}
- T
_{1}
= ma
-------------------
T
_{2}
= 2ma
If the numerical values for T
_{2}
and m are given, then this equation will allow you to solve for the acceleration of the system. Once the acceleration is known, then the tension in cord #1 can also be calculated.
Hanging Masses
Now suppose that the right mass is hanging off a frictionless table and the cord connecting it to the left mass passes over a "massless, frictionless" pulley.
How would this change the freebody diagrams for each mass and their equations of motion?
mass on table
hanging mass
net F
_{x}
= ma
_{x}
T = ma
------
net F
_{y}
= ma
_{y}
- mg = 0
mg - T = ma
Solving these equations simultaneously for the acceleration yields the equation:
T = ma
mg - T = ma
--------------------
mg = 2ma
a = ½g
Once again, if the numerical value for m is given then this equation will allow you to solve for the acceleration of the system. Remember that "g" represents the acceleration due to gravity, 9.8 m/sec
^{2}
. Once the acceleration is known, then the tension in the cord can be calculated.
Refer to the following information for the next six questions.
Obviously, the masses in the previous example do not have to be the same. Solve for the acceleration of the system and the tension in the cord if the hanging object has a mass of 3 kg and the mass on the table is 2 kg.
Write the equation for net F
_{x}
= ma
_{x}
for the 2 kg mass.
Write the equation for net F
_{y}
= ma
_{y}
for the 2 kg mass.
Write the equation for net F
_{x}
= ma
_{x}
for the 3 kg mass.
Write the equation for net F
_{y}
= ma
_{y}
for the 3 kg mass.
What is the acceleration of the system?
What is the tension in the cord?
Atwood Machine
Refer to the following information for the next six questions.
In our third example the two masses are attached to the ends of a single cord that passes over a massless, frictionless pulley suspended from the ceiling. This situation is called an Atwood Machine.
Write the equation for net F
_{y}
= ma
_{y}
for the 2 kg mass.
Write the equation for net F
_{y}
= ma
_{y}
for the 5 kg mass.
Why did we not need to subscript the tension variables?
Why did we not need to write the equations for net F
_{x}
= ma
_{x}
for these two bodies?
What is the acceleration of the system?
What is the tension in the cord?
Related Documents
Lab:
Labs -
Coefficient of Friction
Labs -
Coefficient of Friction
Labs -
Coefficient of Kinetic Friction (pulley, incline, block)
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Falling Coffee Filters
Labs -
Force Table - Force Vectors in Equilibrium
Labs -
Inelastic Collision - Velocity of a Softball
Labs -
Inertial Mass
Labs -
LabPro: Newton's 2nd Law
Labs -
Loop-the-Loop
Labs -
Mass of a Rolling Cart
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Relationship Between Tension in a String and Wave Speed
Labs -
Relationship Between Tension in a String and Wave Speed Along the String
Labs -
Static Equilibrium Lab
Labs -
Static Springs: Hooke's Law
Labs -
Static Springs: Hooke's Law
Labs -
Static Springs: LabPro Data for Hooke's Law
Labs -
Terminal Velocity
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Ball Re-Bounding From a Wall
Labs -
Video Lab: Falling Coffee Filters
Resource Lesson:
RL -
Advanced Gravitational Forces
RL -
Air Resistance
RL -
Air Resistance: Terminal Velocity
RL -
Forces Acting at an Angle
RL -
Freebody Diagrams
RL -
Gravitational Energy Wells
RL -
Inclined Planes
RL -
Inertial vs Gravitational Mass
RL -
Newton's Laws of Motion
RL -
Non-constant Resistance Forces
RL -
Properties of Friction
RL -
Springs and Blocks
RL -
Springs: Hooke's Law
RL -
Static Equilibrium
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Universal Gravitation and Satellites
RL -
Universal Gravitation and Weight
RL -
What is Mass?
RL -
Work and Energy
Worksheet:
APP -
Big Fist
APP -
Family Reunion
APP -
The Antelope
APP -
The Box Seat
APP -
The Jogger
CP -
Action-Reaction #1
CP -
Action-Reaction #2
CP -
Equilibrium on an Inclined Plane
CP -
Falling and Air Resistance
CP -
Force and Acceleration
CP -
Force and Weight
CP -
Force Vectors and the Parallelogram Rule
CP -
Freebody Diagrams
CP -
Gravitational Interactions
CP -
Incline Places: Force Vector Resultants
CP -
Incline Planes - Force Vector Components
CP -
Inertia
CP -
Mobiles: Rotational Equilibrium
CP -
Net Force
CP -
Newton's Law of Motion: Friction
CP -
Static Equilibrium
CP -
Tensions and Equilibrium
NT -
Acceleration
NT -
Air Resistance #1
NT -
An Apple on a Table
NT -
Apex #1
NT -
Apex #2
NT -
Falling Rock
NT -
Falling Spheres
NT -
Friction
NT -
Frictionless Pulley
NT -
Gravitation #1
NT -
Head-on Collisions #1
NT -
Head-on Collisions #2
NT -
Ice Boat
NT -
Rotating Disk
NT -
Sailboats #1
NT -
Sailboats #2
NT -
Scale Reading
NT -
Settling
NT -
Skidding Distances
NT -
Spiral Tube
NT -
Tensile Strength
NT -
Terminal Velocity
NT -
Tug of War #1
NT -
Tug of War #2
NT -
Two-block Systems
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Calculating Force Components
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Combining Kinematics and Dynamics
WS -
Distinguishing 2nd and 3rd Law Forces
WS -
Force vs Displacement Graphs
WS -
Freebody Diagrams #1
WS -
Freebody Diagrams #2
WS -
Freebody Diagrams #3
WS -
Freebody Diagrams #4
WS -
Introduction to Springs
WS -
Kinematics Along With Work/Energy
WS -
Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
WS -
Lab Discussion: Inertial and Gravitational Mass
WS -
net F = ma
WS -
Practice: Vertical Circular Motion
WS -
Ropes and Pulleys in Static Equilibrium
WS -
Standard Model: Particles and Forces
WS -
Static Springs: The Basics
WS -
Vocabulary for Newton's Laws
WS -
Work and Energy Practice: Forces at Angles
TB -
Systems of Bodies (including pulleys)
TB -
Work, Power, Kinetic Energy
PhysicsLAB
Copyright © 1997-2023
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton