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The explanation of this apparently astonishing coincidence is: We defined the center of mass precisely so this is exactly what we would get. Recall that first we defined the momentum of the system:

p CM = j = 1 N d p j d t .

We then concluded that the net external force on the system (if any) changed this momentum:

F = d p CM d t

and then—and here’s the point—we defined an acceleration that would obey Newton’s second law. That is, we demanded that we should be able to write

a = F M

which requires that

a = d 2 d t 2 ( 1 M j = 1 N m j r j ) .

where the quantity inside the parentheses is the center of mass of our system. So, it’s not astonishing that the center of mass obeys Newton’s second law; we defined it so that it would.

Summary

  • An extended object (made up of many objects) has a defined position vector called the center of mass.
  • The center of mass can be thought of, loosely, as the average location of the total mass of the object.
  • The center of mass of an object traces out the trajectory dictated by Newton’s second law, due to the net external force.
  • The internal forces within an extended object cannot alter the momentum of the extended object as a whole.

Conceptual questions

Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How does the explosion affect the motion of the center of mass? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?

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Problems

Three point masses are placed at the corners of a triangle as shown in the figure below.

A right triangle with sides length 3 c m and 4 c m has masses of 100 g at the vertex between the hypotenuse and the 4 c m side, 75 g at the vertex between the hypotenuse and the 3 c m side, and 150 g at the vertex between the the 3 c m side and the 4 c m side.

Find the center of mass of the three-mass system.

With the origin defined to be at the position of the 150-g mass, x CM = −1.23 cm and y CM = 0.69 cm

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Two particles of masses m 1 and m 2 separated by a horizontal distance D are released from the same height h at the same time. Find the vertical position of the center of mass of these two particles at a time before the two particles strike the ground. Assume no air resistance.

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Two particles of masses m 1 and m 2 separated by a horizontal distance D are let go from the same height h at different times. Particle 1 starts at t = 0 , and particle 2 is let go at t = T . Find the vertical position of the center of mass at a time before the first particle strikes the ground. Assume no air resistance.

y CM = { h 2 1 4 g t 2 , t < T h 1 2 g t 2 1 4 g T 2 + 1 2 g t T , t T

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Two particles of masses m 1 and m 2 move uniformly in different circles of radii R 1 and R 2 about origin in the x , y -plane. The x - and y -coordinates of the center of mass and that of particle 1 are given as follows (where length is in meters and t in seconds):
x 1 ( t ) = 4 cos ( 2 t ) , y 1 ( t ) = 4 sin ( 2 t )

and:
x CM ( t ) = 3 cos ( 2 t ) , y CM ( t ) = 3 sin ( 2 t ) .

  1. Find the radius of the circle in which particle 1 moves.
  2. Find the x - and y -coordinates of particle 2 and the radius of the circle this particle moves.
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Two particles of masses m 1 and m 2 move uniformly in different circles of radii R 1 and R 2 about the origin in the x , y -plane. The coordinates of the two particles in meters are given as follows ( z = 0 for both). Here t is in seconds:
x 1 ( t ) = 4 cos ( 2 t ) y 1 ( t ) = 4 sin ( 2 t ) x 2 ( t ) = 2 cos ( 3 t π 2 ) y 2 ( t ) = 2 sin ( 3 t π 2 )

  1. Find the radii of the circles of motion of both particles.
  2. Find the x - and y -coordinates of the center of mass.
  3. Decide if the center of mass moves in a circle by plotting its trajectory.

a. R 1 = 4 m , R 2 = 2 m ; b. X CM = m 1 x 1 + m 2 x 2 m 1 + m 2 , Y CM = m 1 y 1 + m 2 y 2 m 1 + m 2 ; c. yes, with R = 1 m 1 + m 2 16 m 1 2 + 4 m 2 2

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Find the center of mass of a one-meter long rod, made of 50 cm of iron (density 8 g cm 3 ) and 50 cm of aluminum (density 2.7 g cm 3 ).

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Find the center of mass of a rod of length L whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the x -axis with one end at the origin and the other end at x = L , the density is given by ρ ( x ) = ρ 0 + ( ρ 1 ρ 0 ) ( x L ) 2 , where ρ 0 and ρ 1 are constant values.

x c m = 3 4 L ( ρ 1 + ρ 0 ρ 1 + 2 ρ 0 )

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Find the center of mass of a rectangular block of length a and width b that has a nonuniform density such that when the rectangle is placed in the x , y -plane with one corner at the origin and the block placed in the first quadrant with the two edges along the x - and y -axes, the density is given by ρ ( x , y ) = ρ 0 x , where ρ 0 is a constant.

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Find the center of mass of a rectangular material of length a and width b made up of a material of nonuniform density. The density is such that when the rectangle is placed in the xy -plane, the density is given by ρ ( x , y ) = ρ 0 x y .

( 2 a 3 , 2 b 3 )

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A cube of side a is cut out of another cube of side b as shown in the figure below.

A large cube of side b has a cube of side a cut out of its bottom left front corner.

Find the location of the center of mass of the structure. ( Hint: Think of the missing part as a negative mass overlapping a positive mass.)

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Find the center of mass of cone of uniform density that has a radius R at the base, height h , and mass M . Let the origin be at the center of the base of the cone and have + z going through the cone vertex.

( x CM , y CM , z CM ) = ( 0,0 , h / 4 )

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Find the center of mass of a thin wire of mass m and length L bent in a semicircular shape. Let the origin be at the center of the semicircle and have the wire arc from the + x axis, cross the + y axis, and terminate at the − x axis.

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Find the center of mass of a uniform thin semicircular plate of radius R . Let the origin be at the center of the semicircle, the plate arc from the + x axis to the −x axis, and the z axis be perpendicular to the plate.

( x CM , y CM , z CM ) = ( 0 , 4 R / ( 3 π ) , 0 )

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Find the center of mass of a sphere of mass M and radius R and a cylinder of mass m , radius r , and height h arranged as shown below.

Figure a has a sphere on top of a vertical cylinder. Figure b has a sphere centered on top of a horizontal cylinder.

Express your answers in a coordinate system that has the origin at the center of the cylinder.

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Questions & Answers

Hi all, love you all!!!
Giorgi Reply
How does resonance occur
Rahim Reply
what is quantam
pamit Reply
quantum is a division of mechanics
Baje
what is friction
Muhammad Reply
a force act by surface between two bodies whose are always oppose the relative motion .....
Raghav
when two rough bodies are placed in contact and try to slip each other ... than a force act them and it's ippse the relative motion between them
Raghav
thats friction force and roughnes of both bodies is define friction of surface
Raghav
what is a progressive wave
sheriff-deen Reply
What is the wake for therapist
Ife Reply
can u like explain your question with clear detail
Chikamso
who would teach me vectors?
Tintin Reply
what's chemistry
Esther Reply
branch of science dt deals with the study of physical properties of matter and it's particulate nature
Josiah
Good
Daniel
actually
Nathz
Y acctually do u hav ur way of defining it? just bring ur iwn idear
Daniel
well, it deals with the weight of substances and reaction behind them as well as the behavior
Josiah
buh hope Esther, we've answered ur question
Josiah
what's ohms law
CHIJIOKE
ohms law states that, the current flowing through an electric circuit is directly proportional to the potential difference, provided temperature and pressure are kept constant
Josiah
what is sound
James
ohms law states that the resistance of a material is directly proportional to the potential difference between two points on that material, if temperature and other physical conditions become constant
Chikamso
How do I access the MCQ
Abraham Reply
As I think the best is, first select the easiest questions for you .and then you can answer the remaining questions.
lasitha
I mean I'm unable to view it
Abraham
when I click on it, it doesn't respond
Abraham
ohhh,try again and again ,It will be showed
lasitha
okay
Abraham
what is centripetal force
Don Reply
هي قوة ناتجة من الحركة الدائرية ويكون اتجاهها إلى المركز دائماً
meaning of vector quantity
Felix Reply
vector quantity is any quantity that has both magnitude in terms of number (units) and direction in terms of viewing the quantity from an origin using angles (degree) or (NEWS) method
LEWIS
vector quantity is physical quantity has magnitude and direction
vector is a quantity that is use in measuring size of physical properties and their direction
Bitrus
what difference and similarities between work,force,energy and power?
Anes Reply
I need the best answer
Anes
power
mehreen
power
saba
enery is the ability to do work. work is job done, force is a pull or push. power has to do with potential. they belong to different categories which include heat energy, electricity.
Andrew
force refers to a push or pull... energy refers to work done while power is work done per unit time
Shane
mathematically express angular velocity and angular acceleration
Mario Reply
it depends on the direction. an angular velocity will be linear and angular acceleration will be an angle of elevation.
Andrew
The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?
Shaina Reply
Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See [link] .) (a) Calculate the echo times for temperatures of 5.00°C5.00°C and 35.0°C.35.0°C. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty an
Shaina
give a reason why musicians commonly bring their wind instruments to room temperature before playing them.
Shaina
The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0°C,37.0°C, which is the same as body tempe
Shaina
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C10.0°C to 30.0°C30.0°C ? That is, find the ratio of the frequencies at those temperatures.
Shaina
what are vector quantity
Aondover Reply
Quantities that has both magnitude and direction
NNAEMEKA
what is lenses
Rhoda
vector quantities are those physical quantites which have magnitude as well as direction and obey the laws of vector algebra.
Huzaif
electric current has both magnitude and direction but it doesn't obey the laws of vector algebra, hence it is not a vector quantity.
Huzaif
Practice Key Terms 4

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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