Learning Goal: To practice Tactics Box 7.1 Finding the center of gravity. Even though you can locate an object’s cen…
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Question “Learning Goal: To practice Tactics Box 7.1 Finding the center of gravity. Even though you can locate an object’s cen…”
Learning Goal: To practice
Tactics Box 7.1 Finding the center of gravity.
Even though you can locate an object’s center of gravity by
suspending it from a pivot, this is rarely a practical technique.
More often, we would like to calculate the center of gravity of an
object made up of a combination of particles. The following Tactics
Box shows how to find the center of gravity of any number of
particles.
- Choose an origin for your coordinate system. You can choose any
convenient point as the origin. - Determine the coordinates , , , for the particles of mass
, , , respectively. - The x coordinate of the center of gravity is
- Similarly, the y coordinate of the center of gravity
is
the figure, lie on three corners of a square 10.0 on a side. Determine the x
coordinate of each coin, , , and .
separated by commas.
separated by commas.
coordinates and of the center of gravity of the
three coins described in Part A.
centimeters.
Answer
This problem can be solved using 2-D geometry and expressions of the center mass.
Use 2-D geometry to determine the coordinates of center gravity for each coin and use these coordinates to calculate center gravity for the whole system.
2-D Geometry
Any point A on two-dimensional plane can be represented by its coordinate\left( {x,y} \right)
Here $math_tag_1
The center of gravity is an assumed point within the body where all mass is concentrated and the net gravitational force acts.
Complex bodies can be calculated the center of mass by.
x = \frac{{{M_1}{x_1} + {M_2}{x_1} + \cdots + {M_n}{x_n}}}{{{M_1} + {M_2} \cdots + {M_n}}} y = \frac{{{M_1}{y_1} + {M_2}{y_1}...... + {M_n}{y_n}}}{{{M_1} + {M_2}...... + {M_n}}}Here, (x,y)
(A)
Take a look at the diagram.
Select $math_tag_5
Next, you can find the coin at $math_tag_6
{x_C} = 10.0 {\rm{cm}}To purchase coin, visit $math_tag_8
{x_B} = 0.0 {\rm{cm}}To purchase coin at $math_tag_10
{x_A} = 0.0 {\rm{cm}}
(B)
Select $math_tag_5
For coin, go to $math_tag_6
{y_C} = 0.0{\rm{cm}}To purchase coin, visit $math_tag_8
{y_B} = 0.0{\rm{cm}}To purchase coin at $math_tag_10
{y_A} = 10.0{\rm{cm}}
(C)
Let M
Then $math_tag_1
{x_{cg}} = \frac{{M{x_A} + M{x_B} + M{x_C}}}{{M + M + M}},Substitute value for $math_tag_22
\begin{array}{c}\\{x_{cg}} = \frac{{M\left( {0.0 {\rm{cm}}} \right) + M\left( {0.0 {\rm{cm}}} \right) + M\left( {10.0 {\rm{cm}}} \right)}}{{M + M + M}}\\\\ = \frac{{M\left( {10.0 {\rm{cm}}} \right)}}{{3M}}\\\\ = 3.33 {\rm{cm}}\\\end{array}Similarly, y {y_{cg}} = \frac{{M{y_A} + M{y_B} + M{y_C}}}{{M + M + M}},
Substitute value for $math_tag_26
$math_tag_27 Ans: Part A
The values for $math_tag_28
Conclusion
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