show that product of total mass of the system of particles and acceleration at it's center of mass is equal to the vector sum of all the external forces acting on the system of particles? if know correct answer otherwise don't reply!
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Module 6 -- Center of Mass and the motion of a system
Learning Objectives
After the end of this module you should be able to:
Explain why only a net external force can change the translational motion of the center of mass of a system
Determine the motion of the center of mass of a system knowing the external forces acting on the system
Introduction
In this module we will understand the relevance of the concept of the center of mass in the description of the translational motion of a system. By knowing the net force on a system we can determined the acceleration of the system's center of mass; hence the position of the center of mass as a function of time. Applying these results to an extended object, we will justify the point particle approximation that we have been using in the the previous units.
Motion of a Multi Body System
We will start by studying a system of N particles and generalize the results to a single extended object and then to a system of extended objects.
Velocity and Acceleration of the Center of Mass
In this section we will calculate the velocity and the acceleration of the center of mass of a system consisting on N point particles. If the velocity, the acceleration and the mass of the i-th particle is ,
and mi, respectively, and the total mass of the system is M = m1 + m2 + .. + mn, then we have:
Velocity of the CM
[show]Derivation
Acceleration of the CM
[show]Derivation
Total Momentum and Net External Force
Total Momentum
The total linear momentum of the system is equal to that of a single particle of mass M moving with the velocity of the center of mass.
[show]Derivation
Total External Force
The center of mass of a system of particles of mass M moves like an equivalent particle of mass M would move under the influence of the net external force on the system.
[show]Derivation
Review the above derivations by watching two short movies:
[show]1. Only the external forces are relevant (~ 2 min long)
[show]2. Part of Prof. Lewin's video lecture (~ 6 min long)
Remarks
[show]1. Point Particle Behavior of a Rigid Object
[show]2. External Impulse:
[show]3. Zero net external force
Illustrative Examples
[show]Example 1. Walking and running on a slab
[show]Example 2: Unloading a car from a barge
[show]Example 3: Speeding up a boat.
[show]Example 4: Two blocks connected by a spring
[show]Example 5: An explosion in air.
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Course Outline
Introduction
Unit 1 -- Newton's Laws
Unit 2 -- Interactions and Force
Unit 3 -- Applying Newton's Laws
Unit 4 -- Describing Motion
Unit 5 -- Core Models and Planar Dynamics
Unit 6 -- Applying SIM to Problems in Planar Dynamics
Unit 7 -- Momentum and Multi-
Body Systems
Unit 8 -- Mechanical Energy and Work
Unit 9 -- Torque and Rotation About a Fixed Axis
Unit 10 -- Describing Rotational and Translational Motion
Unit 11 -- Angular Momentum
Unit 12 -- Gravitational Orbits
Unit 13 -- Harmonic Oscillation
Unit 14 -- Review
Unit x -- Rotational kinematics
Unit 13 -- New Module Format
System Constituents in Mechanics
point particle
rigid body
massless object
infinitely massive object
Interactions Studied in Mechanics
contact interaction
gravitational interaction
tension interaction
elastic restoring interaction
Hierarchy of Models for Mechanics
Dynamics and Net Force
Momentum and External
Impulse
Rotational Dynamics about a Fixed Axis and Net Torque
Angular Momentum and
External Angular Impulse
about a Single Axis
Mechanical Energy, External
Work, and Internal Non-
Conservative Work
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